Irving  Stringham 


LIPPINCOTT'S 
PRACTICAL  ARITHMETIC 

EMBRACING 

THE   SCIENCE   AND   PRACTICAL   APPLICATIONS 
OF   NUMBERS 


BY 

J.  MORGAN  RAWLINS,  A.M. 
// 

AUTHOR  OF  "LIPPINCOTT'S  ELEMENTARY  ARITHMETIC"  AND  "  LIPPINCOTT'S 
MENTAL  ARITHMETIC" 


PHILADELPHIA 

J.  B.  LIPPINCOTT    COMPANY 


COPYRIGHT,  1899. 

BV 

T.  B.  LIPPINCOTT  COMPANY, 


ELECTROTVPED  AND    PRINTED    BY  J .  B.   LlPPINCOTT   COMPANY,  PHILADELPHIA,   U.  S.  A. 


PREFACE. 


THAT  Arithmetic  is  both  a  science  and  an  art  is  not  only 
generally  conceded,  but  emphatically  affirmed.  One,  there- 
fore, investigating  the  methods  of  instruction  adopted  in 
school-rooms  would  logically  expect  to  find  both  aspects  of 
the  subject  distinctly  put  in  evidence.  What  is  universally 
acknowledged  and  proclaimed  as  essential  and  vital  to  any 
true  system  of  arithmetical  education  the  investigator,  how- 
ever, would  fail  to  find  distinctly  characterizing  either  the 
instructions  given  by  the  average  teacher  or  the  work  required 
of  the  average  pupil.  What  we  mean  is,  that  if  he  saw  any- 
thing notable,  he  would  see  Art  conspicuous  in  the  foreground, 
while  Science,  if  visible  at  all,  sat  mute  far  back.  There 
might  be  seen,  to  be  sure,  remarkable  skill  displayed  in  many 
instances,  and  results  brought  forth  with  surprising  facility ; 
but,  in  it  all,  Science,  that  alone  imparts  life  to  action  and 
informs  the  mind,  would  have  little  or  no  part. 

The  great  error  is  that  arithmetical  exercises  and  problems 
are  too  frequently — nay,  almost  invariably — set  up  like  so 
many  ten-pins,  to  be  knocked  down  by  mechanical  action, 
without  any  inquiry  as  to  the  underlying  and  fundamental 
principles  upon  which  action  is  based.  In  a  word,  the 
schools,  with  little  exception,  are  not  making  the  best  use  of 

800553 


iv  PREFACE 

Arithmetic  as  an  educational  force  by  ignoring  the  fact  that 
it  has  principles  to  be  explained,  induction  and  analysis  to 
explain  them,  and  a  philosophical  reason  for  every  step  neces- 
sary to  be  taken. 

The  text-books  used  may  sometimes  be  seriously  at  fault, 
want  of  time  may  be  an  impediment,  and  other  hindrances 
may  be  numerous ;  but  the  live  and  intelligent  teacher  will 
find  in  the  least  scientific  treatise  food  for  quickening  thought 
and  moulding  the  mental  state.  It  must  be  admitted,  how- 
ever, that  the  teacher,  in  the  conscientious  performance  of  his 
duty,  has  a  difficult  environment,  and  needs  all  the  help  that 
text-books  can  furnish  him. 

The  book  that  we  now  introduce  to  the  public  we  have 
aimed  to  make  what  it  assumes  to  be, — a  Practical  Arithmetic; 
— practical,  not  so  much  by  devising  short  processes  and 
labor-saving  schemes  as  by  laying  a  scientific  foundation  to  be 
studied  and  mastered  as  the  essential  preliminary  to  the  intel- 
ligent and  skilful  use  of  any  device  of  mere  art  ;  practical, 
therefore,  as  a  teacher's  true  assistant,  bringing  to  his  hand  a 
full  supply  of  definitions,  inductive  steps,  illustrations,  prin- 
ciples, analyses,  syntheses,  processes,  rules,  and  suggestions, 
needful  to  him  in  his  high  vocation, — a  vocation  that  is 
highest  when  most  devoted  to  "bright-eyed  Science,"  and 
lowest  when  it  rests  content  with  the  pretentious  and  empty 
forms  of  mere  "  mechanic  art." 

The  text-book,  even  in  its  best  estate,  replete  with  science 
and  art-full,  can  have  little  philosophical  efficiency  except 
when  intelligently  used  as  a  means  to  an  end.  In  the  school- 


PREFACE  V 

room,  where  a  book  is  expected  to  promote  the  high  aims  of 
education,  the  intelligent  use  of  it  must  begin,  if  it  begins  at 
all,  with  the  teacher ;  for  it  is  he  alone  whose  very  office  it  is, 
through  voice  and  action,  to  stir  into  quickening  force  the 
words  of  the  text,  that  otherwise  may  fall  as  good  seed  upon 
sterile  ground.  Every  teacher  ought  to  know — what  every 
pupil  soon  learns — .that  "to  hear  illustrations  and  explana- 
tions from  living  lips  is  a  different  thing  from  struggling 
through  them  on  the  printed  page."  Every  page  is  to  be 
learned,  however, — mastered, — and  made  emphatically  the 
pupil's  own ;  and,  as  a  suggestion  pertinent  here,  we  quote 
the  philosophic  words  of  John  Locke  :  "  The  great  art  to  learn 
much  is  to  undertake  a  little  at  a  time.  " 

The  author  would  gladly  express  his  thanks  to  all  who  in 
any  way  made  contributions  of  help.  To  one  friend,  whose 
devotion  to  the  work  never  faltered,  he  acknowledges  lasting 

obligation. 

J.  M.  R. 


CONTENTS. 


PART    I. 


Definitions 1 

Notation   and    Numera- 
tion      2 

Arabic  Notation 3 

Notation  of  IT.  S.  Money  .  11 

Roman  Notation 13 

Simple  Numbers. 

Addition 15 

Subtraction 24 

Multiplication    ......  36 

Division 50 

Analysis 64 

Indicated  Solutions  ....  65 
General   Principles   of  Di- 
vision    68 

Properties  of  Numbers.  71 

Factoring 73 

Cancellation 77 

Common  Divisors 80 

Common  Dividends  ....  84 

Review 90 

Fractions 91 

Reduction 94 

Addition ...  101 

Subtraction 104 

Multiplication 107 

Division 112 

Complex  Fractions   .    .    .    .  115 

Fractional  Relations .    ...  116 

Review -  118 

Decimal  Fractions    ...  125 

Notation  and  Numeration  .  126 


PAGE 

U.  S.  Money 129 

Reduction 130 

Complex  Decimals    ....  132 

Addition 132 

Subtraction 136 

Multiplication    ......  138 

Division 142 

Short  Processes 144 

Review 149 

Accounts  and  Bills ...  152 

Denominate  Numbers    .  156 

Reduction  Descending ...  157 

Reduction  Ascending  ...  159 

Measures 161 

Of  Length 161 

Of  Surface 163 

Of  Volume 168 

Of  Capacity 174 

Of  Weight 178 

Of  Time 182 

Circular  Measure  ....  184 
Reduction    of   Denominate 

Fractions  (Special)    .    .  187 

Fractional  Relation  ....  189 

Addition 193 

Subtraction 196 

Multiplication 198 

Division 199 

Longitude  and  Time  .   .  200 

Standard  Time 202 

Miscellaneous  Problems  204 

Review 206 

vii 


CONTENTS 


PART    II. 


PAGE 

Percentage 208 

Review  Exercises.   .   .  218 

Commercial  Discount  .    .    .  221 

Gain  and  Loss 223 

Commission 227 

Review 230 

Stocks  and  Bonds 233 

Insurance 240 

Direct  Taxes 243 

Indirect  Taxes 246 

Interest 249 

Six-per-cent.  Method   .    .  253 

Exact  Interest   .       ...  255 

Compound  Interest  .    .    .  259 

Annual  Interest    ....  261 

Promissory  Notes 262 

Partial  Payments 266 

Merchants'  Kule    ....  267 

U.  S.  Kule 269 

Bank  Discount 273 

True  Discount 277 

Review 279 

Exchange 281 

Domestic  Exchange  .    .    .  283 

Ratio  and  Proportion     .  286 

Eule  of  Three 289 

Compound  Proportion     .    .  292 

Cause  and  Effect 295 

Proportionate  Parts  ....  296 

Partnership 298 

Averages 302 

Averaging  or  Equating 

of  Payments    ...  303 

Involution 307 

Evolution 311 

Square  Koot 311 

Cube  Koot H20 

Similar  Figures 326 


PAGE 

Mensuration 329 

Surfaces 330 

Triangles 331 

Parallelograms 332 

Trapezoid 333 

Trapezium 334 

Kegular  Polygon  .  • .    .    .  334 

Circle 335 

Miscellaneous  Problems  336 

Volumes 338 

Prism  and  Cylinder .    .    .  339 

Pyramid  and  Cone    .    .    .  340 

Frustums 341 

Sphere 342 

Circle  and  Largest  Square  343 

Sphere  and  Largest  Cube  344 

General  Review 346 

Appendix 399 

Duodecimals 399 

Metric  System 402 

Foreign  Exchange  ....  408 

Arithmetical  Progression  .  411 

Geometrical  Progression  .  .  412 

Compound  Interest  ....  414 

Annuities 416 

Circulating  Decimals  .  .  .  420 
Greatest  Common  Divisor  of 

Fractions -121 

Least  Common  Dividend  of 

Fractions 421 

The  Thermometer  ....  422 

The  Clock 423 

Work 424 

Averaging  Accounts  .  .  .  425 

Keview 427 

Miscellaneous  Problems  .  .  430 

Table  of  Commercial  Laws  434 


GENERAL  SUGGESTIONS. 


1.  There  is  no  royal  road  to  a  knowledge  of  Arithmetic, 
and  in  this  book  no  attempt  has  been  made  to  preclude  the 
necessity  for  laborious  effort  without  which,  it  has  been  wisely 
said,  life  gives  nothing  to  mortals. 

2.  Self-reliance  is  the  basis  of  action,  and  "  self-activity  is 
the  law  of  growth."     To  render  the  pupil  self-reliant,  self- 
helpful,  and  self-acting  in  the  face  of  difficulties,  is  the  object 
the  teacher  should  keep  steadily  in  view. 

3.  Two  ideas  are  fundamental :    I.  Knowledge  cannot  be 
successfully  built  except  on  knowledge  already  acquired.    The 
lesson  to  be  learned  to-morrow  must  start  its  growth  in  the 
lesson  learned  to-day.     II.  Lessons  assigned  a  class  must  not 
be  made  too  easy  for  some,  nor  too  difficult  for  others.     Care- 
ful judgment  is,   therefore,   required  that  oral  explanations 
and  illustrations  be  neither  too  ample  nor  too  meagre.     The 
least  active  mind  must  be  made  to  understand,  and,  at  the 
same  time,  the  most  active  brain  must  be  required  to  labor. 

4.  In  each  division  of  the  subject,  as  treated,  will  be  found 
inductive  steps,  definitions,  principles,  processes,  explanations, 
rules,  exercises,  and  problems, — all  to  be  thoroughly  learned 
and  intelligently  recited.     Mastery  of  the  exercises  will  give 
facility  in  performing  the  operations  reojiired  by  the  problems. 


x  GENERAL  SUGGESTIONS 

5.  The  solution  of  a  problem  requires  three  distinct  steps : 

I.  The  indication  in  arithmetical  language  of  the  opera- 
tions to  be  performed. 

II.  The  mechanical  performance  of  the  operations  indi- 
cated. 

III.  The  statement  of  the  reasoning  by  which  the  opera- 
tions as  indicated  were  obtained,  and  also  the  elucidation 
of  any  merely  mechanical   step  that  has  been  taken  to 
reach  the  final  result. 

If  in  each  subject  the  introduction,  including  principles, 
processes,  and  explanations,  be  systematically  and  thoroughly 
acquired,  the  exercises  and  problems  that  follow  will  seem  not 
forbidding  obstacles,  but,  as  it  were,  beckoning  friends. 

6.  Too  much  importance  cannot  be  attached  to  the  method 
of  dealing   with   a   problem,  as   pointed   out   above.      The 
frequent  suggestions   made  throughout   the  book  attest  the 
author's  belief  in  the  excellence  of  the  system  proposed.     One 
advantage  is  that  the  first  step — the  indication  in  arithmetical 
language  of  the  work  to  be  done, — really  solves  the  problem,  and 
that  here,  in  many  cases,  the  pupil's  work  may  be  considered 
as  satisfactorily  closed.     Every  teacher  must  determine  for 
himself,  and  every  intelligent  teacher  will  successfully  deter- 
mine, how  far  his  pupils  need  to  work  out  and  recite  the 
details  of  a  solution.     He  must  go  far  enough  to  be  con- 
vinced that  they  have  got  within  them  a  conception  of  the 
truth,  and  are  able  to  declare  it.     But  how  is  this  possible, 
unless  he  recognizes  the  great  fact  that  every  pupil  is  an  indi- 
vidual, has  a  distinct  individuality,  and  is,  as  far  as  possible, 


GENERAL  SUGGESTIONS  xi 

to  be  individually  approached  and  trained,  "  not  for  school, 
but  for  life'7  ? 
To  summarize : 

1.  Do  not  go  too  fast;  hasten  slowly. 

2.  Assign  lessons  with  care,  keeping  in  mind  that  "too 
much  is  not  good." 

3.  Repeat   constantly ;    "  repetition   is   the   mother   of  all 
learning/' 

4.  Require  hard  work ;  "  the  harder  a  pupil  has  worked 
for  what  he  knows  and  can  do,  the  better  for  him." 

5.  Be  methodical,  enthusiastic,  persistent,  and  patient. 

6.  Remember  the  ancient  maxim,  that  "  to  the  boy  is  due 
the  highest  reverence." 


PRACTICAL    ARITHMETIC 


PART    I. 


DEFINITIONS. 

1.  A  Unit  is  a  single  thing  or  one. 

2.  A  Number  is  a  unit  or  a  collection  of  units. 

3.  Arithmetic  is  both  a  Science  and  an  Jr£ :  as  a  science, 
it  investigates  the  principles  of  numbers ;  as  an  art,  it  applies 
those  principles  to  practical  purposes. 

4.  A  Principle  is  a  fundamental  truth  or  ground  of  action. 

5.  A  number  is  Concrete  or  Denominate  when  its  unit 
is  named,  as  in  one  man,  two  books,  three  ships. 

6.  A  number  is  Abstract  when  its  unit  is  not  named,  as 
one,  two,  three. 

When  named,  the  unit  of  a  number  is  one  of  the  things 
expressed  by  the  number,  as  one  tree,  one  man. 

When  not  named,  the  unit  of  a  number  is  simply  one. 

7.  A  Simple  Denominate  number  has  a  single  unit,  as  in 
five  feet.     A  Compound  Denominate  number  has  two  or 
more  related  units,  as  in  three  yards  two  feet  six  inches. 

What  is  the  unit  of  the  concrete  number  three  ships  ?     Of 
the  abstract  number  three  ? 

l 


2  PRACTICAL    ARITHMETIC 

Tell   which   of   the  following   numbers   are  concrete  and 
which  abstract,  and  what  is  the  unit  of  each  : 

1.  Ten  men.  7.  One  apple.  13.  Four  horses. 

2.  Three.  8.  Seven.  14.  Six  wagons. 

3.  Nine  boys.  9.  Five  pounds.  15.  Sixty. 

4.  Eleven  girls.  10.  Fifty-five.  16.  Sixty-seven. 

5.  Twenty -one.  11.  Twenty-nine.  17.  Twenty  ships. 

6.  Seventeen.  12.  Twelve.  18.  Twenty-nine. 

8.  Analysis  (Greek,  taking  apart)  examines  the  separate 
parts  of  a  subject,  or  proposition,  and  their  connection  with 
each  other ;  it  solves  problems  by  a  comparison  of  their  ele- 
ments; it  reasons  from  the  given  number  to  one,  and  then 
from  one  to  the  required  number ;  it  reasons,  also,  from  par- 
ticular instances  to  general  principles. 

9.  Synthesis  (Greek,  putting  together)  unites   separated 
parts,  in  accordance  with  their  obvious  relations. 

10.  A  Rule  is  founded  on  some  principle,  and  is  a  precise 
direction  for  solving  a  problem. 

11.  A  Problem  is  a  practical  question  requiring  a  solution. 

12.  A  Solution  consists  of  a  process  and  an  explanation 
made  by  the  application  of  a  rule  or  by  analysis  and  synthesis. 


NOTATION  AND  NUMERATION. 

1.  Notation  is  the  art  of  writing  numbers. 

2 .  Numeration  is  the  art  of  reading  numbers. 

3.  There  are  three  methods  of  notation  in  common  use : 

1.  The  word  method. 

2.  The  Arabic  or  figure  method. 

3.  The  Roman  or  letter  method. 

4.  The  Arabic  method  employs  the  Arabic  figures  :  1,  2,  3, 
4,  5,  6,  7,  8,  9,  0. 


NOTATION    AND    NUMERATION  3 

5.  The  word  method  names  these  figures  and  expresses 
their  values  as  follows  : 

1,      2,       3,  •     4,       5,      6,       7,         8,        9,         0. 
One,  two,  three,  four,  five,  six,  seven,  eight,  nine,  naught 
(cipher,  zero). 

The  script  forms  are  as  follows  : 

I    -234-51018^0 

These  figures  are  frequently  called  digits  (Latin,  digitus, 
finger) ;  those  preceding  0  are  called  significant  figures. 


ARABIC    NOTATION. 

1.  Each  of  the  first   nine  numbers,  you  perceive,  is  ex- 
pressed by  a  single  digit ;  higher  numbers  are  expressed  by 
combinations  of  the  digits. 

One  prefixed  to  naught  (10)  is  ten. 

2.  Our  system  of  notation  is  "due  to  the  fact  that  we  have 
ten  fingers/7  and  the  basis  of  it  is  the  first  ten  numbers  formed 
into  a  model  group. 

10  is  one  ten,  or  simply  ten  (Latin,  "  decem"). 

11  is  eleven  (Gothic,  "am,  one;  lif,  ten"),  one  and  ten. 

12  is  twelve  (Gothic,  "  tva,  two;  lif,  ten"),  two  and  ten. 

13  is  thirteen,  three  and  ten. 

14  is  fourteen,  four  and  ten. 

15  is  fifteen,  five  and  ten. 

16  is  sixteen,  six  and  ten. 

17  is  seventeen,  seven  and  ten. 

18  is  eighteen,  eight  and  ten. 

19  is  nineteen,  nine  and  ten. 

20  is  twenty  (teen  becomes  ty). 


4  PEACTICAL    ARITHMETIC 

3.  The  numeral  names  that  precede  "  teen"  follow  "  ty"  and 
a  hyphen  (-),  as  follows :  21  is  twenty-one,  that  is,  twice  ten 
and  one ;  30  is  thirty ;  31  is  thirty-one ;  40  is  forty ;  42  is 
forty-two. 

4.  Any  significant  figure,  located  as  four  in  40,  has  its 
value  increased  ten-fold  and  denotes  tens. 

Locate  5  thus,  and  name  the  number ;  also  6,  7,  8,  9. 

The  value  of  a  significant  figure  in  units7  place  is  called  its 
Simple  Value. 

The  value  of  a  significant  figure  otherwise  placed  is  called 
its  Local  Value. 

5.  44  is  four  tens  and  four  units,  or  forty- four. 

Which  four  has  the  increased  or  Local  Value  1  Which  has 
only  its  Simple  Value  ?  •* 

99  is  ninety- nine,  and  is  the  largest  number  that  can  be 
expressed  with  two  figures. 

6.  100  is  ten  tens,  or  one  hundred. 
200  is  twenty  tens,  or  two  hundreds. 
300  is  thirty  tens,  or  three  hundreds. 

Any  significant  figure  thus  located  expresses  hundreds. 

444  is  four  hundreds,  four  tens,  four  units,  or  four  hundred 
forty-four. 

404  is  thus  read  :  "  four  hundred  four,"  not  "  four  hundred 
and  four."  The  naught  (0)  indicates  the  absence  of  tens. 

7.  Again,  404,  or  any  three  digits  thus  written  together, 
constitute  a  period  with  units  in  the  first  place,  tens  in  the 
second  place,  and  hundreds  in  the  third  place.     The  left  hand 
period  may  contain  but  one  or  two  digits. 

8.  Ten  units  grouped  make  a  single  ten-group.     Ten  ten- 
groups   make   a   single   hundred-group.     Ten  one-hundred- 
groups  make  one  thousand,  written  1000. 

9.  Any  single  thing  is  a  unit ;  a  single  ten-group  may,  there- 
fore, be  considered  a  unit ;  so  also,  a  single  hundred -group. 


NOTATION    AND    NUMERATION  5 

1O.  On  this  principle  a  digit  in  the  first  place  denotes  units 
of  the  first  order ;  in  the  second  place,  units  of  the  second 
order  ;  in  the  third  place,  units  of  the  third  order,  etc. 

PRINCIPLES. 

1.  The  first   nine  numbers   are   expressed   by  the  nine 
digits  (1,  2,  3,  etc.),  taken  singly. 

2.  Numbers  above  nine  are  expressed  by  combining  the 
digits  and  giving  them  local  values. 

3.  Naught  (O)  has  no  value,  but  is  used  to  fill  a  vacant 
place  and  to  fix  the  values  of  the  significant  figures. 

4.  Local  value  increases  from  right  to  left,  ten  units  of 
any  order  making  one  unit  of  the  next  higher  order. 

EXERCISES. 

NOTE. — Pupils  should  be  carefully  drilled  in  giving  the  digits  their 
correct  forms.  Ill-formed  figures  often  lead  to  erroneous  results. 

1.  Write  the  Arabic  numerals. 

2.  Write  their  names. 

3.  Write  the  significant  figures. 

4.  Write  figures  enough  to  make  a  period. 

5.  Write  a  period  and  indicate  the  absence  of  tens  and 
units. 

6.  Write  two  thousand  three  hundred  seventy-five. 

7.  How   many   places   have   you  written?     How  many 
orders  ?     How  many  periods  ? 

8.  How  many  units  of  any  order  make  one  unit  of  the 
next  higher  order  ? 

9.  Our  system  of  notation  puts  how  many  units  in  a  group  ? 

10.  What  is  a  unit?     When  may  ten  or  a  hundred  be 
considered  a  unit? 

11.  Write  units  of  the  fourth  order,  and  show  the  absence 
of  units  of  the  first,  second,  and  third  orders. 


6 


PRACTICAL   ARITHMETIC 


12.  Express  1898  in  words,  remembering  what  was  said 
about  "  and." 

NUMERATION   TABLE. 

1.  Places,  orders,  and  periods  may  be  carried  on  indefinitely 
from  right  to  left. 

2.  The  whole  subject  may  now  be  concisely  presented  in 
tabular  form.     The  first  six  periods  are  as  follows : 


NAMES  OF 
PERIODS. 


0 

1 

r^    oT 

c 

CO 

NAMES  OP 

•il 

.2 

O 

PLACES.  ' 

3     £2       «3 

rs  § 

172    »T 
*-     C 
•*?  .0 

1  S 

PLACES  AND  ] 
ORDERS.      -1 

(  oo  Hundred 
-j  oo  Ten-quac 
1  oo  Quadrilli 

9   S     C 

£  i  .2 

§    g? 
W  H  H 

888, 

^  a 
2  S 

i^?. 

§    g  r 
W  H  P 

8   8 

V  Y  

PERIODS.                GTH. 

5TH. 

4TH. 

VI 

§ 

c 

ci 

Q 

1 

oT 

e 

cc 

c 

Q 

C 

rC 

c 

oT 

i 

H3 

1 

c 
o 

1 

H3 

| 

^ 
1 

C 

W 

1 

1 

C 
W 

1 

c 
H 

8 

v^_ 

8 

8, 

^_-* 

8 

v,^  . 

8 

8, 

^.^ 

3D 

2D 

1ST. 


3.  The  periods  are  separated  from  each  other  by  commas. 

4.  The  periods  from  the  first   to  the  twenty-second  are 
named  as  follows : 


1.  Units. 

2.  Thousands. 

3.  Millions. 

4.  Billions. 

5.  Trillions. 

6.  Quadrillions. 

7.  Quintillions. 


8.  Sextillions. 

9.  Septillions. 

10.  Octillions. 

11.  Nonillions. 

12.  Decillions. 

13.  Undecillions. 

14.  Duodecillions. 

15.  Tredecillions. 


16.  Quatuordecillions. 

17.  Quindecillions. 

18.  Sexdecillions. 

19.  Septendecillions. 

20.  Octodecillions. 

21.  Novendecillions. 

22.  Vigintillions. 


NOTATION    AND    NUMEKATION  7 

EXERCISES. 

1.  Write  five,  fifty-five,  five  hundred  fifty-five,  and  state 
what  each  five  expresses. 

2.  Write  a  number  consisting  of  four  digits,  and  point 
off  the  first  period  with  a  comma ;  also,  read  the  number. 

3.  Write  a  number  consisting  of  two  full  periods ;  name 
the  periods,  and  read  the  number. 

4.  In  876,  of  what  order  and  place  is  each  figure? 

5.  In  writing  nine  hundred  seven,  how  will  you  express 
the  tens  ?     Write  the  number. 

6.  Write  a  number  consisting  of  three  periods ;  name  the 
periods,  and  read  the  number. 

7.  Write  numbers  consisting  respectively  of  four  periods, 
five  periods,  six  periods,  and  read  each  of  the  numbers. 

8.  Write  a  number  with  a  significant  figure  located  in  the 
seventh  place.     What  will  occupy  the  other  six  places  ? 

9.  If  a  number  to  be  written  omits  a  period,  or  an  order, 
or  a  place,  what  must  in  all  cases  supply  the  vacancy  ? 

10.  Write  a  number  with  7  in  the  sixth  place,  5  in  the 
fourth  place,  and  2  in  the  first  place. 

1 1 .  Write  a  number  with  five  full  periods  and  one  partial 
period. 

12.  Point  off  into  periods  and  read  405651320. 

13.  In  the  preceding  number,  how  many  units  of  the  first  N 
order?     How  many  of  the  eighth  order?     How  many  ten- 
thousands  ?     How  manv  ten-millions  ? 


RULE  FOR  NUMERATION. 

1.  Begin   at   the  right   and   mark  off  the  number   into 
periods. 

2.  Begin  at  the  left,  read  each  period  separately,  naming 
each  period  except  that  of  units. 


PRACTICAL  ARITHMETIC 


Copy  and  read 


89. 
134. 
946. 
1664. 
5790. 
83405. 
624151. 
731052. 
8000000. 

10.  763303454. 

11.  900058798. 

12.  100100001. 


1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 


EXERCISES. 

13.  571320179. 

14.  35627003. 

15.  1000000000. 

16.  4321078654. 

17.  4141414441. 

18.  62340007313. 

19.  141662223143. 

20.  700706831455. 

21.  31671240630231. 

22.  1987000634596521612912. 

23.  1234567891011121314151. 

24.  6171819202122232425262. 


1 .  "Write  the  number  five  hundred  twenty-three. 

Process.  Explanation. 

ANALYSIS. — Five  hundred  twenty-three  means  five  hun- 
523        dreds,  two  tens,  three  units,  represented  by  the  digits  5,  2, 

and  3. 

SYNTHESIS. — We,  therefore,  write  3  in  the  first  place,  which  is  the 
place  of  simple  units,  2  in  the  second  place,  which  is  the  place  of  tens, 
and  5  in  the  third  place,  which  is  the  place  of  hundreds. 

2.  Write  six  hundred  thirty-seven  thousand  one  hundred  six. 

Process.  Explanation. 

ANALYSIS. — Since  the  given  number  contains  637  thou- 
637,106  sands  and  106  units,  there  are  in  it  two  full  periods. 

SYNTHESIS. — We,  therefore,  write  the  digits  of  the  thou- 
sands, 637,  as  the  second  period,  and  the  digits  of  the  units,  106,  as  the 
first  period. 


NOTATION    AND    NUMERATION  9 

3.  Express  in  figures  : 

1.  Five  hundred  sixty-four. 

2.  Seven  hundred  fifty-  nine. 

3.  Four  thousand  eighty-one. 

4.  One  thousand  two  hundred. 

5.  Twenty-five  thousand  seven. 

6.  Forty-one  thousand  nineteen. 

7.  Six  thousand  six  hundred  six. 

8.  One  hundred  thirty-one  thousand. 

9.  Sixty-five  thousand  four  hundred  seventy  -nine. 

10.  One  million  five  hundred  three  thousand  five  hun- 

dred ninety-three. 

11.  Ninety-one  million  three  hundred  forty-five  thousand. 

12.  Twelve  thousand  nine  hundred  seventy-eight. 

13.  Thirty-  one  billion  three  hundred  thirteen  million  six 

hundred  seventy-two  thousand  four  hundred  eleven. 

14.  One  hundred  sixty-four  million  eighteen  thousand. 

15.  One  hundred  fifteen  quadrillion  four  hundred  forty- 

four  trillion  five  hundred  three  billion  four  million 
two  hundred  fifty  thousand  one. 

DECIMAL   PARTS    OP    A   UNIT. 

1.  By  placing  a  mark  (.)  called  the  decimal  point  after 
units  of  the  first  order,  the  numeration  and  notation  table  is 
extended  to  express  parts  of  a  unit,  on  the  decimal  scale  : 


5.5  5  5 

The  above  number  is  thus  read  :  "  five  and  five  hundred 
fifty-five  thousandths." 


10 


PRACTICAL   ARITHMETIC 


2.  The  decimal  point  (.)  is  always  read  "  and." 
6.7  is  read  "  six  and  seven  tenths." 

7.89  is  read  "  seven  and  eighty-nine  hundredths." 
.005   is   read  "five  thousandths."     The  naughts  are 
only  recognized  as  giving  local  value  to  5. 

1.234  is  read  "one  and  two  hundred  thirty-four  thou- 
sandths. 

3.  The  decimal  point  never  acts  as  a  period. 


EXERCISES. 

1.  Read :    .08,  .75,  .006,  3.079. 

2.  Write  :  Twenty-seven  hundredths. 

Eleven  thousandths. 

One  and  four  hundred  six  thousandths. 

Thirteen  and  twenty- five  thousandths. 

3.  Read:    17.6,1.76,  .196,  .144. 

4.  Write  :  Five  hundred  six  thousandths. 

Five  hundred  and  six  thousandths. 

5.  Read  :    325.72,  325,  32.5,  .072. 

6.  Write  :  Five  hundred  four  thousandths. 

Five  hundred  and  four  hundredths. 

7.  Read  :    6.050,  7.200,  872.003,  .409. 

8.  Write  :  Seven  tenths  four  thousandths. 

Nine  and  seventy  thousandths. 

9.  Express  "  and"  by  a  sign. 

10.  What  word  interprets  the  decimal  point. 

11.  What  is  the  difference  between  a  decimal  point  and  a 
period  ? 

12.  Form  a  number  by  writing  the  digits  six,  seven,  eight, 
nine,  and  zero,  in  their  natural  order ;  then  place  the  decimal 
point  in  all  the  different  positions  you  can  ;  finally  read,  in 
succession,  the  different  numbers  you  have  thus  formed. 


NOTATION    AND    NUMERATION  11 

13.  State  the  effect  of  moving  the  point  one  place  to  the 
right ;  one  place  to  the  left. 

14.  How  many  fold  does  a  removal  one  place  increase  or 
diminish  the  value  expressed  ? 


UNITED    STATES   MONEY. 

1.  The   currency  of  the   United   States   has   the   decimal 

system. 

Table. 

10  mills      make  1  cent. 
10  cents      make  1  dime. 
10  dimes    make  1  dollar. 
10  dollars  make  1  eagle. 

2.  $  is  the  dollar  sign,  and,  prefixed  to  an  abstract  number, 
renders  it  concrete  :  10  becomes  $10,  read  "  ten  dollars." 

3.  The  dollar  is  the  Unit,  and  the  decimal  point  is  invaria- 
bly placed  between  the  dollars  and  dimes  of  any  sum  of 
money  :  $5.60  is  read  "  five  dollars  and  sixty  cents,"  or  "  five 
dollars  and  six  dimes." 

EXERCISES. 

1.  Express  in  figures  nine  dollars  and  twenty-five  cents 
six  mills. 

Process.  Explanation. 

ANALYSTS. — Given :    nine  dollars,    twenty-five   cents,   six 
$9.256      mills.     Twenty-five  cents  are  two  dimes  and  five  cents. 

The  dollar  is  the  unit. 

SYNTHESIS. — Write  the  dollar  sign,  9,  and  a  point ;  and  after  the  point 
2  dimes,  5  cents,  and  6  mills,  in  their  natural  order. 

2.  Express  in  figures  thirty-one  dollars  and  nine  cents  five 
mills. 


12  PEACTICAL   ARITHMETIC 

Process.  Explanation. 

ANALYSIS. — Given:    thirty-one  dollars,   no  dimes,  nine 
$31.095        cents,  five  mills. 

PRINCIPLE. — 0  supplies  a  vacant  place. 

SYNTHESIS. — Write  dollar  sign,  31,  and  a  point;  0  in  dimes'  place; 
and  9  cents  and  5  mills  in  their  order. 


3.  Write  six  dollars  and  eighty-five  cents. 

4.  Read  $2.235,  $202.025,  $112.25. 

5.  Write  five  hundred  dollars  and  nine  cents  five  mills. 

6.  Write  two  thousand  dollars. 

7.  Write  forty  dollars  and  four  cents. 

8.  Read  $313112.13,  $20000.32. 

9.  Write  twelve  dollars  and  five  cents  six  mills. 

10.  Write  six  thousand  one  dollars  and  one  mill. 

11.  Write  seven  million  dollars  and  seventy-seven  cents. 

12.  Read  $.05,  $.03,  $.62,  $.70. 

13.  Copy  and  read  the  following  : 


1.  $8.53. 

5.   $236.06. 

9.  $796.844. 

2.  $13.75. 

6.  $20000. 

10.    $.16. 

3.  $39.05. 

7.  $2104.083. 

11.    $.057. 

4.  $49.34. 

8.  $6001.102. 

12.  $12.500. 

14.  Write  the  following : 

1.  Eight  dollars  and  fifty  cents. 

2.  Two  hundred  two  dollars  and  two  cents  five  mills. 

3.  Five  dollars. 

4.  Five  hundred  dollars. 

5.  One  hundred  twelve  dollars  and  twenty-five  cents. 

6.  Four  dollars  and  eighty-seven  cents. 

7.  Ninety-seven  cents  eight  mills. 

8.  Six  hundred  twenty  dollars  and  nine  cents. 

9.  Twelve  million  seven  hundred  thousand  dollars. 


NOTATION   AND   NUMERATION 

10.  Three  thousand  ten  dollars  and  fifty  cents. 

11.  Seventy  dollars  and  ten  cents. 

12.  Six  million  dollars  and  eighty  cents. 

13.  Four  cents.     Ten  cents.     Nine  mills. 


13 


ROMAN   NOTATION. 

1.  This  system  of  notation  employs  seven  capital  letters. 


I.  denotes  one,  1. 
V.  denotes  five,  5. 
X.  denotes  ten,  10. 
L.  denotes  fifty,  50. 


Table. 

C.  denotes  one  hundred,  100. 

D.  denotes  five  hundred,  500. 
M.  denotes  one  thousand,         1,000. 
M.  denotes  one  million,     1,000,000. 


2.  All  other  numbers  are  expressed  by  combining  or  repeat- 
ing these  letters : 


I.  .  . 

1. 

XIV.     . 

....  14. 

C  100. 

II..  . 

2. 

XV.  .    . 

....  15. 

CCCC.,orCD.     .    400. 

III.  . 

...       .3. 

XVI.     . 

....  16. 

D  500. 

IV.    . 

4. 

XVII.  . 

....  17. 

DCCCC.,or  CM.    900. 

V.  .   . 

5. 

XVIII 

18 

Ml  000 

VI.    . 

6. 

XIX.     . 

....  19. 

MD  1500. 

VII.  . 

7. 

XX.  .    . 

....  20. 

MDCLXV.  .    .    .  1665. 

VIII. 

8. 

XXI.     . 

.    .  'L  .  21. 

MDCCXLIX.  .    .  1749. 

IX.    . 

9. 

XXX.   . 

....  30. 

MDCCCLXXIX,  1879. 

X.  . 

10. 

XL.   .    . 

....  40. 

V.    5,000. 

XI. 

.        .    .  11. 

L.  .    .    . 

....  50. 

L  50,000. 

XII.  . 

12. 

LX.   .    . 

....  60. 

C  100,000. 

XIII. 

13. 

XC.    .   . 

.   .   .   .90. 

M  1,000,000. 

From  the  repetitions  and  combinations  observable  above, 
we  derive  the  following 


14  PRACTICAL   ARITHMETIC 

PRINCIPLES. 

1.  Repeating  I.,  X.,  C.,  or  M.  repeats  its  value. 

XX.  denotes  20;    CO.  denotes  200.     V.,  L.,  and  D.  cannot  be  thus 
repeated. 

2.  Prefixing  I.,  X.,  or  O.   to   a  letter  of  greater  value 
diminishes  that  value  by  L,  X.,  or  C. 

3.  Afllxing  L,  V.,  X.,  L.,  C.,  or  D.  to  a  letter  of  greater 
value  increases  that  value  by  L,  V.,  X.,  L.,  C.,  or  D. 

4.  Inserting  L,  X.,  or  C.  between  two  letters,*  each  of 
greater  value,  diminishes  the  united  value  of  the  two  by 
I.,  X.,  or  O. 

*  The  first  of  the  two  must  not  be  of  less  value  than  the  second. 
XIV.,  not  VIX.,  denotes  14;  XIX.  denotes  19. 

5.  A  bar  placed  over  a  letter,  except  I.,  increases  its 
value  a  thousand-fold. 

C.  denotes  100,000. 

6.  IIII.   is   sometimes   used   instead  of  IV.,   as   on   the 
dials  of  clocks  and  watches.    4OO  may  be  expressed  by 
COCO,  or  by  CD. 

EXERCISES. 
1.  Read  the  following  combinations : 

1.  XV.  10.  XLV.  19.  DCCXC. 

2.  IV.  11.  XCIX.  20.  MXXIX. 

3.  XIV.  12.  LXV.  21.  VDLV. 

4.  XXIV.  13.  CIX.  22.  DLDC. 

5.  XIX.  14.  CXI.  23.  CCXDVL 

6.  XXXIX.  15.  XCI.  24.  VIII. 

7.  XXXIII.  16.  DCXC.  25.  CCXC. 

8.  XXIX.  17.  CCCXXXIX.  26.  CXLIX. 

9.  XLIX.  18.  DCCXXXIV.  27.  HMD. 


28.  LXXDCCCXCIX.    29.  MDXCVDCCCLXIV. 


NOTATION    AND    NUMEKATION 


15 


2.  Write  in  Roman  characters  the  following 


1. 
2. 
3. 
4. 
5. 
6. 


15. 

36. 
87. 
56. 
49. 
99. 


7.  1050. 

8.  5010. 

9.  789. 
10.  1898. 


11.       18. 

21. 

27. 

12.       42. 

22. 

81. 

13.       66. 

23. 

95. 

14.       86. 

24. 

40. 

15.       63. 

25. 

45. 

16.     100. 

26. 

534. 

17.  3600. 

27. 

5000. 

18.     587. 

28. 

436. 

19.     207. 

29. 

999. 

30.  76,959. 


20.  8004. 

3.  Which  of  these  are  correct  expressions  and  which  in- 
correct ? 

DD. 

CCC. 

XCC. 

MMXL. 

XLIX. 

CIXXVII. 


W. 

LV. 

XIL. 

LXIX. 

XLX. 

XCIX. 


VXX. 

VLC. 

VDC. 

CIXYIIX. 

LXXXVIIL 

DMCC. 


REVIEW'. 


1.  Define  the  following  terms  : 

1.  Unit. 

2.  Number. 

3.  Arithmetic. 

4.  Principle. 

5.  Concrete  number. 

6.  Abstract  number. 

7.  Analysis. 

8.  Synthesis. 

9.  Eule. 
10.  Problem. 


11.  Solution. 

12.  Notation. 

13.  Numeration. 

14.  Word  method. 

15.  Arabic  method. 

16.  Roman  method. 

17.  Simple  value. 

18.  Local  value. 

19.  Zero. 

20.  Period. 


16  PRACTICAL  ARITHMETIC 

21.  Decimal  point.  25.  Dime. 

22.  United  States  Money.  26.  Dollar. 

23.  Mill.  27.  Eagle. 

24.  Cent.  28.  Significant  figures. 

2.  Repeat  the  four  principles  of  notation. 

3.  Name  the  periods  from  the  1st  to  the  22d. 

4.  Repeat  the  five  principles  of  the  Roman  notation. 

5.  Repeat  the  rule  for  numeration. 


ADDITION. 

INDUCTIVE   STEPS. 

1.  How  many  units  are  5  units  and  3  units?     2  tens  and 
7  tens  ?     4  thousands  and  6  thousands  ? 

2.  A  certain  field  has  7  acres,  and  an  adjoining  field  8 
acres.     How  many  acres  in  both  fields  ? 

Process.  Explanation. 

7  acres  Since  one  field  contains  7  acres  and  the  other  8  acres,  the 

8  two  fields  contain  7  acres  and  8  acres,  which  are  15  acres, 

acres 


15  acres          ®.  If  on  one  shelf  there  are  9  books  and  on 
another  shelf  5  books,  how  many  books  are  on 
both  shelves  ? 

"Write  and  explain  the  process. 

4.  How  many  pounds  are  8  pounds  and  6  pounds  ? 

Write  and  explain. 

5.  The  process  of  thus  uniting  quantities  in  a  single  quan- 
tity is  called  adding. 

6.  What  is  the  unit  of  8  pounds?     Of  6  pounds?     Of  14 
pounds  ? 

7.  Can  quantities  having  like  units  be  added  ? 


ADDITION  17 

8.  Add  6  pounds  and  5  dollars.    Can  you  show  a  process? 
If  you  can,  is  your  result  11  pounds  or  11  dollars? 

9.  What,  then,  does  addition  require  as  to  the  units  to  be 
added  ? 

10.  What  does  addition  require  as  to  the  unit  of  the  result 
or  sum  ? 

11.  Numbers  having  like  units  are  called  Like  Numbers. 

DEFINITIONS. 

1.  Addition  is  the  process  of  finding  the  sum  of  two  or  more 
like  numbers.     The  sum  is,  therefore,  the  result  of  addition. 

2.  A  Sign  indicates  some  process  or  condition.     The  sign 
of  addition  is  an  upright  cross,  -{-.     It  is  read  "  plus." 

3.  The  Sign  of  Equality  is  two  short  horizontal  lines,  =. 
It  is  read  "  equals,"  or  "  is  equal  to."     3  -f  2  =  5,  is  read  "  3 
plus  2  equals  5."     3  +  2  =  5,  being  an  expression  of  equality, 
is  called  an  Equation. 

PRINCIPLES. 

1.  Only  like  numbers  and  orders  can  be  added. 

2.  The  numbers  added  and  their  sum  are  like  numbers. 

EXERCISES. 
1.  Find  the  sum  of  120,  331,  and  478. 

Process.  Explanation. 

120  ANALYSIS. — There  are  three  numbers  to  be  added,  each  con- 

oo-i        taining  units,  tens,  and  hundreds. 
^  PRINCIPLE. — Only  like  orders  can  be  added. 

SYNTHESIS. — Hence  we  write  the  numbers  with  the  units' 
929        figures  (0,  1,  8)  in  the  first  column  on  the  right,  the  tens'  figures 
(2,  3,  7)  in  the  second  column,  and  the  hundreds'  figures  (1,  3, 
4)  in  the  third  column. 

The  sum  of  the  first  column  is  9  units ;  the  sum  of  the  second  column 
is  12  tens  =  1  hundred  -[-  2  tens.  We  write  the  2  tens,  and  add  the  1 
hundred  to  the  hundreds'  column,  making  9  hundreds.  Hence  the  sum 
required  is  929. 

2 


18  PRACTICAL   ARITHMETIC 

2.  Find  the  sum  of  25,  206,  and  9837. 

Process.  Explanation. 

25  ANALYSIS. — 

206  25  =  2  tens  -f  5  units. 

9  837  206  =  2  hundreds  -f  0  tens  -f  6  units. 

9,837  =  9  thousands  -f  8  hundreds  4-  3  tens  -4-  7  units. 

-i  r\  A£»Q 

9  PRINCIPLE. — Only  like  orders  can  be  added. 

SYNTHESIS. — We,  therefore,  write  the  units'  figures  (5,  6,  7)  in  the  first 
column;  the  tens'  figures  (2,  0,  3)  in  the  second  column;  the  hundreds' 
figures  (2  and  8)  in  the  third  column ;  and  the  9  thousands  alone  in  the 
fourth  place.  Adding  the  first  column  we  have  18  units  =  1  ten  and  8 
units.  We  add  the  1  ten  to  the  tens'  column  and  have  6  tens.  Adding 
the  third  column  we  have  10  hundreds  =  1  thousand  and  0  hundreds.  We 
say  finally  1  thousand  -j-  9  thousand  =  10  thousand. 

RULE   FOB   ADDITION. 

1.  See  that  the  numbers  to  be  added  are  like  numbers. 

2.  "Write  units  of  the  same  order  in  the  same  column. 

3.  Begin  at  units'   column,  and   find  the   sum  of  each 
column  separately. 

4.  Write  the  units  of  a  sum,  but  add  the  tens  with  the 
next  column. 

5.  "Write  the  entire  sum  of  the  last  column. 

EXERCISES. 
1.  Find  the  sum  of: 

(1.)  (2.)  (3.)  (4.)  (5.) 
234  134  712  473  535 
365  542  314  321  213 


(7.)         (8.)          (9.) 

$6.10  $31.12  $231.25 

$2.11  $41.23  $542.30 

$1.34  $20.44  $210.44 


ADDITION  19 

(10.)         (11.)         (12.)  (13.) 

4134       2460       3782  469 

8104      3782      1856  7206 

3910      3673      1916  39 

45       418      3061  6 


2.  What  is  the  sum  of  2213,  1123,  3201,  2112? 

3.  What  is  the  sum  of: 

1.  3210  -f-  2136  +  3752  +  2331  ? 

2.  3561  -|-  5103  +  6385  +  5632? 

3.  73,250  +  3102  +  16,287  +  1210  +  7542? 

4.  50,673  +  520  +  16,302  -f  2531  +  7204? 

5.  154,632  +  54,231  +  16,302*  +  2120  +  8023? 

4.  Find  the  sum  of: 

(1.)  (2.)  (3.)  (4.)  (5.)  (6.) 

1.  888  +  777  +  666  +  555  +  543  +  735  =  ? 

2.  444  +  333  +  222  +  111  +  210  +  141  =? 

3.  000  +  999  +  234  +  423  +  924  +  287  =  ? 

4.  578  +  287  +  342  +  760  +  553  +  765  =  ? 

5.  504  +  167  +  359  +  578  +  751  +  432  =  ? 

6.  105  +  483  +  142  +  263  +  351  +  109  =  ? 

Add  the  foregoing  both  vertically  and  horizontally. 

5.  What  is  the  sum  of  37  +  375  +  3754  +  37,546  +  64 
+  645  +  4573  -f  57,373? 

6.  Add  $317.50,  $610.10,  $514.085,  $6.16. 

7.  What   is   the   sum   of  four   hundred   sixty-two,   three 
thousand  two  hundred  fourteen,  seventy-nine  thousand  six 
hundred  fifty-nine,  two  hundred  eighty-four? 

8.  What  is  the  sum  of  eighteen  dollars  and  five  cents,  fifty- 
one  dollars,  fifty-one  cents,  ten  dollars  and  ten  cents,  eighteen 
dollars  and  twenty-four  cents,  thirty-five  dollars? 


20  PRACTICAL   ARITHMETIC 

9.  Write  the  following  numbers  with  Arabic  numerals 
and  find  their  sum:  DCCCCXXXVL,  MDXVL, 
MMMMCCIV.,  CLIV.,  XCVIL,  CLXIX. 

10.  4682  +  19,783  +  100  +  6402  +  178  +  19  =  ? 

PROBLEMS. 

Let  the  pupil  first  indicate  the  solution  of  each  problem  by  using  the 
signs,  -(-  and  =. 

1.  A.  owns  345  sheep,  B.  owns  295,  C.  owns  436,  and  D. 
owns  524.     How  many  sheep  do  all  own  ? 

Process  Indicated. 

345  -j-  295  -1-  486  -f  524  =  number  of  sheep  required. 

Process.  Explanation. 

345  ANALYSIS.— There  are  four  flocks  of  sheep :  A.'s  =  345. 

295  B-'s  =  295' 

C.'s=436. 
D.'s  ==  524. 

524 

In  each  number  the  unit  is  1  sheep ;  hence  the  numbers 


1600  are  &&e  and  may  be  added. 

SYNTHESIS.— 345  +  295  -f  436  -f  524  =  1600.     Hence 
all  own  1600  sheep. 

2.  How  many  acres  are  in  three  fields,  containing  respect- 
ively 23  acres,  34  acres,  and  38  acres  ? 

3.  A  man  bought  a  horse  for  $250,  a  carriage  for  $175,  a 
harness  for  $74.50,  a  whip  for  $1.25,  a  carriage  blanket  for 
$3.45.     What  did  he  pay  for  all  ? 

4.  A.  bought  7590  pounds  of  pea  coal,  3765  pounds  of 
nut  coal,  6834  pounds  of  stove  coal,  and  2505  pounds  of 
bituminous  coal.     How  much  coal  did  he  purchase? 

5.  In  a  primary   school   there  are  386  children   in   first 
grade,  258  in  second  grade,  237  in  third  grade,  and  184  in 
fourth  grade.     How  many  pupils  in  the  four  grades  ? 


ADDITION  21 

6.  Spain  has  an  area  of  195,773  square  miles;  France, 
204,091;  Switzerland,  15,922;  Italy,  112,622.     How  great 
is  the  area  of  the  four  countries  ? 

7.  The  battle-ship  "Oregon"  sailed  from  San  Francisco 
to  Callao,  4,012  miles;  from  Callao  to  Sandy  Point,  2,666 
miles ;  from  Sandy  Point  to  Eio,  2,228  miles ;  from  Rio  to 
Bahia,  745  miles  ;  from  Bahia  to  Barbadoes,  2550  miles  ;  from 
Barbadoes  to  St.  Thomas,  346  miles ;  from  St.  Thomas  to  Key 
West,  1040  miles.    Find  the  total  number  of  miles  she  sailed. 

8.  The  monthly  pay  of  a  major-general  in  the  United 
States  army  is  $625 ;  of  a  brigadier-general,  $458.33 ;  of  a 
colonel,  $291.67;  of  a  lieutenant-colonel,  $250;  of  a  major, 
$208.33;  of  a  captain,  mounted,  $166.67;  of  a  captain,  not 
mounted,  $150;   of  a  chaplain,  $125.     Find  total  monthly 
pay  of  the  eight  officers. 

9.  In  1897  the  organized  military  strength  of  the  State 
of  New  York  was  13,894  men ;  of  Pennsylvania,  8521 ;  of 
Illinois,  6260;  of  Ohio,  6004;  of  Massachusetts,  5154;  of 
New  Jersey,  4297  ;  of  California,  3909 ;  of  Georgia,  4450 ; 
of  South  Carolina,  3127;   of  Texas,  3023.     What  was  the 
entire  military  strength  of  the  ten  States  in  1897? 

10.  In  1890  the  population  of  Cincinnati  was  216,239 ;  of 
Cleveland,  92,829  ;  of  Toledo,  31,584  ;  of  Columbus,  31,274 ; 
of  Dayton,  30,473.     How  many  inhabitants  had  these  cities 
altogether  in  1890? 

11.  A  merchant  received  money  for  goods  as  follows :  On 
Monday,  $357.15;    on   Tuesday,  $463.87;    on    Wednesday, 
$279.19;   on  Thursday,  $318.67;   on  Friday,  $687.27;   on 
Saturday,  $348.48.     Find  the  total  receipts. 

12.  A  builder  bought  a  lot  for  $650,  built  upon  it  a  house 
costing  $5845,  a  barn  and  carriage-house  costing  $1075.50; 
he  paid  for  fencing  $215.75,  for  grading  $87.50.     For  what 
must  he  sell  the  property  to  gain  $640  ? 


22  PRACTICAL  ARITHMETIC 

13.  The  provinces  of  Cuba,  with  the  population  of  each, 
are  as  follows : 

Province.  White.  Colored. 

Havana 344,417  107,511 

PinardelRio.     .     .     .  167,160  58,731 

Matanzas    .....  153,169  116,401 

Santa  Clara     ....  249,345  109,777 

Puerto  Principe   .     .     .  54,232  13,557 

Santiago  de  Cuba      .     .  157,980  114,339 

Find  the  total  population  of  Cuba. 

14.  The  land  forces  of  Japan  are  as  follows  :  infantry,  fifty- 
six   thousand   thirty-seven ;    cavalry,   VDCCLX ;    artillery, 
seven  thousand  818 ;   engineers  and  train,  IVCCCXXVI. 
What  is  the  total  land  force  ? 

15.  January  has  31  days,  February  28,  March  31,  April 
30,  May  3t,  June  30,  July  31,  August  31,  September  30, 
October  31,  November  30,  December  31.     How  many  days 
in  a  year  ? 

16.  If  a  school  session  closes  on  the  29th  of  June  and 
opens  again  on  the  10th  of  September,  how  many  days'  vaca- 
tion will  there  be  ? 

17.  What  are  the  expenses  of  a  factory  for  a  year,  if  the 
manager  receives  $1850,  the  engineer  $850,  the  fireman  $650, 
the  bookkeeper  $800,  the  fuel  costs  $1600,  the  raw  material 
$111,110,  and  the  pay-roll  of  the  other  employees  amounts 
to  $55,000  ? 

18.  Add  2,  6,  8,  7,  1,  2,  8,  5,  3,  2,  8,  9. 

19.  Add  IV.,  VIL,  II.,  V.,  V.,  II,  IX.,  VI.,  II.,  VIII., 
VII.,  V. 

20.  A  capitalist  made  the  following  deposits  in  a  bank : 
August  4,  1897,  $484.50;  August  7,  $985.25;  August  10, 
$436.75.    In  a  second  bank  as  follows  :  August  14,  $2657.76  ; 


ADDITION  23 

August  18,  $1386.25;   August  22,  $2096.65.     How  much 
did  he  deposit  in  each  bank  ?     How  much  in  both  banks  ? 


(1.) 

(2.) 

(3.) 

(4-) 

21.  7651 

5005 

10,475 

$453.48 

8923 

4567 

72,482 

4,938.78 

4554 

2299 

46,552 

85,473.89 

5421 

9900 

62,651 

3,457.96 

6432 

8877 

62,272 

835.47 

9888 

7788 

67,286 

53.49 

8797 

6655 

40,025 

9.87 

5032 

5566 

82,827 

82.75 

8060 

4433 

40,050 

875.39 

2134 

3344 

24,165 

48.34 

(5.) 

(6.) 

(7-) 

(8.) 

8410 

6546 

4828 

595 

9836 

3210 

3424 

579 

984 

785 

293 

8574 

543 

156 

788 

3250 

9758 

5634 

2763 

386 

8574 

7654 

5612 

984 

451 

696 

942 

5849 

876 

321 

397 

6546 

7864 

3288 

5945 

429 

5849 

2188 

8020 

451 

762 

785 

694 

8765 

321 

564 

432 

5634 

3250 

7688 

6131 

543 

8765 

5861 

9876 

762 

688 

978 

750 

3210 

642 

643 

976 

3288 

24  PRACTICAL  ARITHMETIC 

(9.)  (10.)  (11.) 

.85  $901.09  .3789 

463.27  91.85  .7398 

39.99  387.24  4.217 

1.58  19,877.46  3.95 

6,598.86  19.90  45.007 

9,005.79  104.99  4.256 

95,783.04  3,972.87  3.520 

2,469.98  79,841.24  23.3 

956.83  18.72  29.317 

14,816.00  3,120.50  343.28 

3,947.25  14.12               1899. 


REVIEW. 

1.  Define  the  following  terms  : 

1.  Like  numbers.  5.  Sign  of  Addition. 

2.  Unlike  numbers.  6.  Equation. 

3.  Addition.  7.  Indicated  process. 

4.  Sum.  8.  Process. 

2.  Repeat  the  principles  of  Addition. 

3.  Repeat  the  rule  for  Addition. 

4.  Invent   five  problems  in  Addition  and   indicate  their 
solution. 


SUBTRACTION. 

INDUCTIVE   STEPS. 

1.  How  many  are  6  units  less  3  units  ?    7  tens  less  5  tens  ? 
8  millions  less  4  millions  ? 

2.  If  you  have  $9  and  spend  $5,  how  many  dollars  do 
you  retain? 


SUBTRACTION  25 

Process.  Solution. 

$9  If  I  have  $9  and  spend  $5,  I  retain  the  difference  between 

AK  $9  and  $5,  which  is  $4. 

$4  Is  that  explanation  analytical  or  synthetical  ? 

3.  $5  -f  $4  =  how  many  dollars? 

4.  Was  it  analysis  or  synthesis  that  gave  you  the  $9  ? 

5.  Does  the  synthesis,  then,  prove  the  correctness  of  the 
analysis  ? 

6.  Robert  is  10  years  of  age  and  Richard  is  8.     What  is 
the  difference  of  their  ages  ? 

Write  and  explain  the  process.     Prove  the  correctness  of  the  result. 

7.  There  were  7  bunches  of  ripe  grapes  on  a  vine;  a  fox 
took  2  bunches.     How  many  bunches  remained  ? 

8.  Have   you   been   finding   the  difference  between  like 
numbers  ? 

9.  Finding  the  difference  between  two  numbers  is  called 
Subtracting. 

10.  What  is  the  difference  between  6  horses  and  3  sheep  ? 

11.  Subtraction  of  numbers  makes  what  requirement  as  to 
their  units  ? 

DEFINITIONS. 

1.  Subtraction   is   the   process  of  finding   the   difference 
between  two  like  numbers. 

2.  The  greater  number  is  called  the  Minuend;    the  less 
number  is  called  the  Subtrahend;    the  result  is  called  the 
Difference  or  Remainder. 

3.  The  Sign  of  Subtraction  is  a  short  horizontal  line,  — , 
called  minus  (less),  and  is  always  placed  after  the  minuend 
and  before  the  subtrahend. 

7  —  5  =  2  is  read  "  7  minus  5  equals  2."     The  form, 
7  —  5  =  2,  is  called  what? 


26  PRACTICAL   ARITHMETIC 

PRINCIPLES. 

1.  Only  like  numbers  and  orders  can  be  subtracted. 

2.  Subtrahend  -f-  Remainder  =  Minuend. 

1.  From  54  subtract  33. 

Process.  Explanation. 

54  The  minuend,  54  =  5  tens  -|-  4  units ; 

00  the    subtrahend,    33  =  8  tens  -f-  3  units. 

2  tens  -f  1  unit  =  21. 

Z*  \ 

PRINCIPLE. — Only  like  orders  can  be  subtracted. 

We  therefore  write  the  3  units  under  the  4  units  and  the  3  tens  under 
the  5  tens.  We  then  say  "  4  units  —  3  units  =  1  unit ;  5  tens  —  3  tens 
=  2  tens.  Hence  the  remainder  is  21." 

Proof. 

PRINCIPLE. — The  subtrahend  -f-  the  remainder  =  the  minuend. 
33  +  21  =  54. 

2.  From  469  subtract  327. 

Process.  Explanation. 

469  469  =  4  hundreds  -(-  6  tens  -|-  9  units. 

007  327  =  3  hundreds  +  2  tens  +  7  units. 

1  hundred   -f  4  tens  -f-  2  units  =  142. 

PRINCIPLE. — Only  like  orders  can  be  subtracted. 

We  therefore  write  the  4,  6,  and  9  of  the  minuend,  and  under  them  the 
3,  2,  and  7  of  the  subtrahend,  with  units  under  units,  tens  under  tens,  and 
hundreds  under  hundreds.  We  now  say  "  9  units  —  7  units  =  2  units; 
6  tens  —  2  tens  =  4  tens  ;  4  hundreds  —  3  hundreds  =  1  hundred.  Hence 
the  difference  is  142."  Show  proof. 

EXERCISES. 

Copy,  subtract,  explain,  prove : 

(1.)           (2.)           (3.)      .     (4.)  (5.)  (6.) 

824         569         997         965  896  8953 

413         245         743         752  544  3420 


SUBTRACTION  27 


(70 

(8.) 

(9.) 

(10.) 

$59.86 

$75.39 

$56.89 

$52.90 

$34.24 

$40.30 

$45.76 

$31.50 

(11.) 

(12.) 

(13.) 

(14.) 

62,979 

98,316 

945.791 

$798.945 

30,825 

71,004 

523.150 

$653.620 

PROBLEMS. 

NOTE. — Let  the  pupil  first  indicate  the  solution  of  each  problem  by 
using  the  minus  sign,  — . 

1.  An  army  went  into  battle  with  6878  men,  and  came  out 
with  only  4345  men.     How  many  men  were  missing? 

Process  Indicated. 
6878  men  —  4345  men  —  the  number  missing. 

Process.  Explanation. 

6878  !•  Since  the  army  went  into  battle  with  6878  men  and 

4345  came  out  with   only  4345,   the  number  missing  was  6878 

. minus  4345. 

2533  2.  Since  the  unit  of  both  the  numbers  is  one  man,  the 

numbers  are  like  and  can  be  subtracted,  the  less  from  the 
greater;  units  from  units,  tens  from  tens,  etc.  Therefore  we  say  "  8  uniti 
—  5  units  =  3  units,  7  tens  —  4  tens  =  3  tens,  8  hundreds  —  3  hundreds 
=  5  hundreds,  6  thousands  —  4  thousands  =  2  thousands.  Hence  the 
number  of  men  missing  was  2533." 

Proof. 

PRINCIPLE. — Subtrahend  - -j-  Remainder  =  Minuend. 
4345  +  2533  =  6878. 

2.  A  grain  dealer,  having  7890  bushels  of  wheat,  sold  6370 
bushels.     How  many  bushels  had  he  remaining  ? 


28  PRACTICAL   ARITHMETIC 

Process  Indicated. 

7890  bushels  —  6370  bushels  =  the  bushels  remaining. 

3.  Watches  were  invented  at  Nuremburg  in  1477.     How 
many  years  ago  ? 

4.  If  I  borrow  $6798,  and  afterwards  pay  $3534,  how 
much  do  I  still  owe? 

5.  Under  a  call  for  volunteers,  California's  quota  was  3237 
men;  Arkansas's  quota,  2025  men.     Find  the  difference? 

6.  The  population  of  Spain  in  1820  was  about  11,000,000 ; 
at  present  (1899)  it  is  17,550,216.     Find  the  increase. 

7.  The  exports  of  the  United  States  from  the  Philippine 
Islands  last  year  amounted  to  $4,982,857 ;  their  imports,  to 
$162,446.     Find  the  excess  of  the  exports  over  the  imports. 

8.  The  population  of  Havana  is   198,720,  of  Santiago, 
71,300.     Find  the  difference. 

9.  The  telescope  was  invented  in  1610.     How  many  years 
between  that  date  and  1899  ? 

10.  Harvey  discovered  the  circulation  of  the  blood  in  1619. 
How  many  years  after  the  invention  of  the  telescope  ? 

CHIEF   DIFFICULTY   OF   SUBTRACTION. 
1.  From  594  take  368. 

Process.  Explanation. 

594  ANALYTIC  AND  SYNTHETIC. 

368  594  =  5  hundreds  -f  9  tens  -f  4  units. 

368  =  3  hundreds  -f  6  tens  -f-  8  units. 

The  difficulty  is  that  8  units  cannot  he  taken  from  4 
units.  But  one  of  the  9  tens  =  10  units  ;  10  units  -f-  4 
units  =  14  units.  Hence  we  write  : 

{QfvS 
594  =  &  hundreds  +  8  tens  +  14  units  1     Subtract. 
368  =  3  hundreds  -f  6  tens  +    8  units  / 

594  ing  we  have  2  hundreds  -f-  2  tens  -f-    6  units  =  226. 


SUBTKACTION  29 

2.  What  is  the  first  principle  of  subtraction  ? 

3.  On  what  principle  does  the  proof  depend? 

4.  From  703  take  549. 

Process.  Explanation. 

e  9  is  703  =  7  hundreds  -f  0  tens  -f-  3  units 

703  549  _  5  hundreds  +  4  tens  -f  9  units 

549 


The  difficulty  is  that  we  cannot  take  9  units  from  3  units, 
nor  4  tens  from  o  tens.     But  one  of  the  7  hundreds  =  10  tens ; 

one  of  the  10  tens  =  10  units  ;  10  units  -j-  3  units  =  13  units.     Hence  we 

may  write 

703  =  6  hundreds  +  9  tens  +  13  units  j     gubtracti       we 
549  =  5  hundreds  -f  4  tens  -f    9  units  J 


have  1  hundred   -f-  5  tens  -[-    4  units  =  154. 

5.  Give  the  principles  of  subtraction  and  prove  the  work. 

6.  From  367.280  take  298.356. 

Process.  Explanation. 

367.280  367.280  =  367  units  +  280  thousandths  1         gub 

298.356  298  356  =  298  units  +  356  thousandths  / 

68.924  tracting  we  have       68  units  -f  924  thousandths  =  68.924. 

7.  Where  must  the  point  always  be  placed  in  the  remainder  ? 


RULE   FOR   SUBTRACTION. 

1.  See    that    the    numbers    to    be    subtracted    are    like 
numbers. 

2.  Write  the  subtrahend  under  the  minuend,  units  under 
units,  etc. 

3.  Beginning  at  the  right,  subtract  each   lower   figure 
from  the  one  above  it. 

4.  When  necessary,  increase  the  upper  figure  by  1O  and 
diminish  by  1  the  next  upper  figure  on  the  left. 


30  PKACTICAL   ARITHMETIC 

EXERCISES. 
1.  Copy,  subtract,  explain,  prove : 


(1.)      (20 

864   1095 
559    867 

(3.) 

937 
645 

(4.)      (5.) 

865   2537 
593    658 

(6.) 

954 

893 

(7.)      (8.) 
2957   -2794 
1038   2406 

(9.) 

3908 
2609 

(10.)     (11.) 
4002   8923 
3962   2095 

(12.) 
9114 
6983 

(13.)     (14.) 

$35.56  $40.19 
$32.49  $38.02 

(15.) 

$58.19 

(16.)     (17.) 

$82.99  $53.44 
$58.03  $19.78 

(18.) 

$6.12 
$5.125 

(19.) 

618.724    9 
529.728    8 

(20.) 

,651,782 
,241,509 

(21.) 

69,503.48 

38,298.75 

(22.) 

8888.88 
7890.10 

2.  What  is  the  value  of: 
1.  81,214  —  53,467? 
2.  104,321  —  58,461  ? 
3.  831,408  —  337,529? 
4.  740,037  —  357,320? 

5.   862,493  - 
6.   998,765  — 
7.  9,327,325  — 
8.  4,986,384  — 

729,603? 
567,890? 
3,586,143? 
2,998,796? 

PROBLEMS. 

1.  If  a  man  owes  $97.66  and  pays  $70.89,  how  much  does 
he  then  owe  ? 

Process  Indicated. 

$97,66  —  $77.89  =  how  much  he  then  owes. 


SUBTKACTION  31 

Process.  Explanation. 

$97  66  Since  he  owes  $97.66,  and  pays  $70.89,  lie  still  owes  the 

r-n'oq  difl'erence  between  $97.66  and  $70.89. 

Since  the  numbers  have  the  same  unit,  one  dollar,  they 
$26.77  are  like  numbers,  and  their  difference  can  be  found.     It  is 

$26.77. 

NOTE. — In  each  problem  let  the  process  be  indicated  first,  and  then 
performed  and  explained. 

2.  A  man  bought  some  land  for  $8765,  and  sold  it  for 
$1 0,890.     What  was  his  gain  ? 

3.  The  first  line  of  telegraph  was  established  in  the  United 
States  in  1844.     How  long  ago? 

4.  In   1890    the   population   of  the   United   States   was 
62,622,250,  and  in  1840  it  was  17,063,353.     How  much  did 
it  increase  in  the  50  years  ? 

5.  A  man  was  born  in  1785  :  what  was  his  age  in  1830? 

6.  How  old  was  George  Washington  at  the  time  of  his 
death?     He  was  born  in  1732,  and  died  in  1799. 

7.  29,400  feet  is  the  greatest  depth  of  water  measured. 
37,000  feet  is  the  greatest  height  reached  by  a  balloon.     Find 
by  how  much  the  greatest  height  reached  exceeds  the  greatest 
depth  reached. 

8.  The   displacement   of   the   battle-ship    "Alabama"   is 
11,525;    of  the   cruiser   "  Charleston,"    3730.      How   much 
does  the  displacement  of  the  "  Alabama"  exceed  that  of  the 
"  Charleston"  ? 

9.  The  estimated  population  of  the  United  States  in  1800 
was  5,308,483 ;  in  1898  it  was  74,500,000.     Find  the  growth 
in  population  in  the  98  years. 

10.  How  many  dollars  must  be  added  to  $4872  to  make 
$8021 ? 

11.  How  many  dollars  increased  by  $74,015  make  a  million 
dollars? 


32  PRACTICAL  ARITHMETIC 

12.  What  number  must   be  taken  from   $6412  to  leave 
$5366  ? 

13.  Find  the  value  of  $8.052  —  $3.687. 

14.  John  has  $20.19  and  James  has  $40.      How  much 
more  money  has  James  than  John  ? 

15.  A  bankrupt  has  $6456  assets,  and  owes  $33,860.    How 
much  more  does  he  owe  than  he  can  pay  ? 

16.  Mt.    Everest   is   29,062    feet   high;    Mt.  Whitney   is 
14,900  feet  high.     How  much  higher  is  the  former  than  the 
latter? 

17.  The  height  of  Mt.  Cenis,  an  Alpine  peak,  is  11,792 
feet ;  the  height  of  the  pass  over  it  is  6884  feet.     How  much 
higher  is  the  mountain  than  the  pass  ? 

18.  At  an  election  3245  persons  voted,  and  the  candidate 
elected  received  1808  votes.     How  many  did  the  defeated 
candidate  receive  ? 

19.  Benjamin  Franklin  died  in  1790,  and  was  84  years  old 
at  his  death.     When  was  he  born  ? 

20.  Subtract  MMMIX.  from  LXVIIXI. 

ADDITION  AND  SUBTRACTION  IN  COMBINATION. 

EXERCISES. 

1.  What  is  the  value  of  20,324  -f  4756  —  13,186? 
Process.  Explanation. 

20  324  ANALYTIC  AND  SYNTHETIC. 

4  755  The  sign  -f  signifies  that  I  must  add  20,324  and  4756. 

Adding,  the  sum  is  25,080. 

25,080  The  sign  —  signifies  that  I  must  subtract  from  that  sum 

13,186  13,186.     Subtracting,  the  remainder  is  11,894. 

11,894  2    what  ig  the  yalue  of . 

1.  23,732  —  9478  -f  9273  ? 

2.  25,657  -f  10,898  —  2597  ? 


SUBTRACTION  33 

3.  20,201  —  9022  +  2002  ? 

4.  132,571  —  90,798  +  78,318  ? 

5.  $238.70  —  $53.36  +  $22.27  ? 

A  Parenthesis,  (  ),  or  Vinculum,  ,  Indicates  that 

all  the  quantities  it  incloses  are  to  be  considered  as  a  single 
quantity ;  as,  (2  +  5  +  10  +  13),  or  2  +  5  +  10  +  13. 

2.  What  is  an  Equation  ? 

3.  Prove  the  following  equations  to  be  correct : 

First  perform  the  operations  indicated  within  the  parentheses. 

1.  40  —  (2  +  5  +  10  +  13)  =  10. 

2.  (355  +  637  +  403)  —  977  —  418. 

3.  2543  —  504  +  600  +  725  =  714. 

4.  10,000  —  (275  +  220  +  35  +  3675)  =  5795. 

5.  (300  +  100  +  95  +  60  +  125)  —  125  +  25  +  40 

—  490. 

4.  Complete  the  following  partial  equations  : 

1.  (350,000  +  225,100  +  4000  +  96,000)  —  450,120 


2.  23,191,876  —  3,204,313  +  434,495  =  ? 

3.  (367  +  875  +  1012)  —  423  +  912  =  ? 

4.  (36  +  200  +  150)  —  331  =  ? 


PROBLEMS. 

NOTE. — The  indicating  of  a  solution  often  facilitates  the  completing  of 
it.  The  pupil  should  be  faithfully  drilled  in  the  use  of  signs  to  indicate 
the  actual  solution  to  be  made. 

1.  Mr.  A.  gave  his  note  for  $6000.  He  paid  at  one  time 
$3586  and  at  another  time  $2000.  How  much  remained  to 
be  paid  ? 

Process  Indicated. 

$6000  —  ($3586  -f  $2000)  =  debt  remaining. 


34  PRACTICAL   ARITHMETIC 

Process.  Explanation. 

Paid $3586  Since  he  paid  $3586   at  one  time  and 

Paid 2000  $2000  at  another  time,  he  paid  at  both  times 

$3586  +  $2000,  or  $5586. 

Total  paid    .        5586  Since  his  note,  or  debt,  was  $6000,  he 

still  owes  the  difference  between  $6000  and 
Note $6000  15586,  or  $414. 

Paid 5586 

2.  A  man  finished  a  journey  of 

Balance  due,  $414  972  miles  in  3  days;  the  first  day 
he  travelled  398  miles;  the  second  day,  409  miles.  How 
many  miles  did  he  travel  the  third  day  ? 

Process  Indicated. 

972  miles  —  (398  miles  -f  409  miles)  =  miles  travelled  third  day. 

3.  A  produce  dealer  had  in  bank  $6032,  and  checked  out 
on  one  day  $2360,  and  on  the  next  day,  $2307.     How  much 
had  he  left  in  bank  ? 

4.  A  cattle  dealer  had  982  cattle,  bought  621  more,  lost 
by  disease  32,  sold  416.     How  many  remained? 

5.  How  much  does  the  sum  of  3694  and  5005  exceed  the 
difference  of  10,532  and  3903? 

6.  In  1880  there  were  16,120  Indians  and  75,025  Chinese 
in  California.     How  many  were  there  of  both,  and  how  many 
more  Chinese  than  Indians  ? 

7.  A.  sells  a  house  to  B.  for  $3486 ;  B.  sells  it  to  C.  at  a 
gain  of  $360  ;  C.  sells  it  to  D.  at  a  loss  of  $285.     What  does 
D.  pay  for  the  house  ? 

8.  I  have  a  yearly  income  of  $10,000.     I  pay  $450  for 
rent,  $230  for  fuel,  $50  for  medical  attendance,  and  $4786  for 
all  my  other  expenses.     What  have  I  saved  at  the  end  of  the 
year  ? 

9.  A  tract   of   land   containing   2753    acres  was   divided 
among  four  persons,  A.,  B.,  C.,  D.     A.'s  share  was  679  acres, 


SUBTRACTION  35 

B.'s  was  47  acres  more,  C.  had  75  acres  less  than  B.,  and  D. 
had  the  remainder.    What  were  the  shares  of  B.,  C.,  and  D.  ? 

10.  A  man  has  an  income  of  $1845 ;  he  spends  $645  for 
board,  $456  for  clothing,  and  $297  for  other  expenses.    What 
has  he  saved  at  the  end  of  the  year  ? 

11.  A  man  deposits  in  bank  $2374.    At  one  time  he  draws 
out  $897,  at  another,  $543,  and  at  a  third  time,  $689.     How 
much  has  he  remaining  in  bank? 

12.  I  bought  24  shares  of  bank  stock  for  $2863,  and  paid  a 
broker  $22  for  purchasing  the  same;  afterwards  sold  it  for 
$3000.     What  was  my  profit? 

13.  A  farmer  invests  $18,975  as  follows :  in  land,  $11,893  ; 
in  horses,  $1575;    in   mules,  $4297;    in   stock,  $937;   the 
remainder  in  tools.     How  much  did  he  expend  for  tools? 

14.  A  man  bought   three  houses;    for  the  first  he  gave 
$3585 ;  for  the  second,  $5260 ;  for  the  third,  as  much  as  for 
the  other  two.     He  sold  them  all  for  $15,280.     Did  he  gain 
or  lose,  and  how  much  ? 

15.  From  the  sum  of  874  and  398  subtract  their  difference. 

16.  I  have  a  bin  that  holds  936  bushels.     I  put  into  it  383 
bushels,  and  again  457  bushels.     How  much  more  will  the 
bin  hold? 

17.  Some  excursionists   made   a  journey   that   cost  them 
$492.97.    Railroad  fares  cost  $203.26  ;  hack  hire  cost  $48.36  ; 
steamboat  fare  cost  $72.46.     The  remainder  was  expended  for 
food.     What  did  their  food  cost? 

18.  A  merchant  bought  silk  for  $486,  muslin  for  $286, 
linen  for  $346,  and  sold  the  whole  for  $1200.     How  much 
did  he  gain  ? 

19.  An  estate  worth  $23,460  was  bequeathed  to  a  wife  and 
two  children.     The  widow  received  $7820 ;  the  son  received 
$3400  less ;  and  the  daughter,  the  balance.    Find  the  daughter's 
share. 


36  PKACTICAL  ARITHMETIC 

REVIEW. 

1.  Define  the  following  terms  : 

1.  Minuend.  6.  Analysis. 

2.  Subtrahend.  7.  Synthesis. 

3.  Difference  8.  Solution. 

4.  Remainder.  9.  Parenthesis. 

5.  Minus.  10.  Vinculum. 

2.  Repeat  the  principles  of  Subtraction. 

3.  Repeat  the  rule  for  Subtraction. 

4.  Invent  five  problems  involving  both  Addition  and  Sub- 
traction, and  indicate  the  process  of  solution. 


MULTIPLICATION. 

INDUCTIVE   STEPS. 

1.  How  many  are  3  +  3  ? 

2.  How  many  are  3  -f  3  +  3  ? 

3.  How  many  are  two  times  3  ? 

4.  How  many  are  three  times  3  ? 

5.  3  -f  3  -f-  3  =  9,  and  3  times  3  =  9.    The  first  operation  is 
addition  ;  the  second  is  multiplication.    Which  is  the  shorter  ? 

6.  Is  Multiplication  a  short  kind  of  addition  ? 

7.  What  will  4  apples  cost  at  2  cents  apiece  ? 

Process.  Explanation. 

ADDITION.     MULTIPLICATION.          Since  1  apple  costs  2  cents,  4  apples 

will  cost  4  times  2  cents,  which  are  8 

2  cents  cents. 

In   the   preceding    process,   is 
2cente  2   cents    concrete?      Is    8   cents 

2  cents  J^ concrete?     Is  the  4  concrete  or 

8  cents  8  cents  abstract? 


MULTIPLICATION  37 

8.  What  i's  the  cost  of  5  yards  of  ribbon  at  10  cents  a  yard  ? 

Write  and  explain  the  process. 

In  the  preceding  process,  did  you  write  5  yards  or  simply  5  ? 
What  kind  of  number  is  5  ? 

You  took  10  cents  five  times ;  hence  5  is  called  the  Multi- 
plier, and  the  process  is  called  Multiplication. 

DEFINITIONS. 

1.  Multiplication  is  a  short  process  of  finding  the  sum  of 
two  or  more  equal  numbers ;  or  of  taking  a  number  as  many 
times  as  there  are  units  in  another  number. 

2.  The    Multiplicand    is    the    number    to    be    taken    or 
repeated. 

3.  The  Multiplier  is  the  number  which  shows  how  many 
times  the  multiplicand  is  to  be  taken  or  repeated. 

4.  The  Product  is  the  result  of  the  multiplication. 

5.  The  Multiplicand  and  Multiplier  are  called  Factors  of 
the  product. 

6.  The  Sign  of  Multiplication  is  an  oblique  cross,  X- 

5  x  4  ==  20,  is  read  "  5  times  4  are  20"  ;  or,  "  5  multiplied 
by  4  equals  20." 

7.  What  are  the  factors  of  20  ? 

Is  4x5  —  5x4a  correct  equation  ? 

Proof. 

***** 
***** 
***** 
***** 

The  20  stars  as  arranged  equal  5  stars  in  a  line  taken  4  times, 
or  4  stars  in  a  column  taken  5  times. 

Hence  the  product  is  the  same  in  whatever  order  the  factors 
are  taken. 


38  PRACTICAL   ARITHMETIC 

8.  What  is  a  parenthesis  and  what  does  it  signify?  A 
vinculum  ? 

134  — -  (9  +  6)  X  3  signifies  that  you  must  take  9  +  6 
three  times  and  subtract  the  result  from  134. 

Process. 

(9  +  6)  X  3  =  45;  134  —  45  =  89. 


PRINCIPLES. 

1.  The  multiplier  must  be  considered  an  abstract  number. 

2.  The  product  and  multiplicand  are  like  numbers. 

3.  Either  factor  may  be  taken  as  the  multiplier. 

EXERCISES. 

FOR  ANALYTIC  AND  SYNTHETIC  EXPLANATION. 
1.  What  is  the  product  of  347  multiplied  by  3? 

Process.    Profess.    Process.  Explanation. 

347  IST  PROCESS. — "We  say,  "Apro- 

o  duct  is  the  result  of  multiplication. 

Since  multiplication  is  a  short  process 


347  21  of  adding  equal  numbers,  we  can  find 

347               1 2  347  ^e  Pr°duct  by  addition ;   adding,  we 

347           9  3  have  1041-" 

2o    PROCESS. — We    say,    "  Since 


1041  1041  1041  347  must  be  taken  3  times,  each  order 

of  units  must  be  taken  3  times.     3 

times  7  units  =  21  units ;  3  times  4  tens  =  12  tens  ;  3  times  3  hundreds  = 
9  hundreds;  adding,  we  have  1041." 

3D  PROCESS. — The  shortest  process  is  generally  the  best  in  practice. 
We  say,  "3  times  7  =  21  ;  we  reserve  the  2;  3  times  4  =  12,  12  and  2 
reserved  are  14 ;  we  reserve  the  1 ;  3  times  3  =  9,  9  and  1  reserved  —  10. 
Hence  the  product  is  1041." 

Proof. 

The  first  and  second  processes  are  proof  of  the  accuracy  of  the  third. 


MULTIPLICATION  39 

2.  Complete,  explain,  and  prove  the  following  : 

(1.)  (2.)  (3.)  (4.)  (5.) 

365  674  756  327  408 

24365 


(6) 

(70 

(8.) 

(9.) 

(10.) 

$5.09 

$7.95 

$12.75 

$55.06 

$43.60 

5 

8 

7 

9 

12 

3.  How  many  places  in  the  product  must  be  pointed  off  for 
cents  ? 

4.  Multiply  : 

1.  8692  by  8.  6.  13,896  by  3. 

2.  5328  by  7.  7.  52,209  by  5. 

3.  10,318  by  5.  8.  68,387  by  7. 

4.  7289  by  9.  9.  79,588  by  4. 

5.  17,345  by  4.  10.  91,983  by  9. 

5.  Find  the  value  of: 

1.  420  x  9.  5.     40,527  X  4. 

2.  9059  x  2.  6.  305,238  X  5. 

3.  78,059  X  3.  7.     40,597  X  6. 

4.  1,790,478  x  7.  8.  910,362  X  8. 

The  parts  of  an  equation,  right  and  left  of  the  sign  of 
equality,  are  called  its  members. 

6.  Find  a  second  member  for  each  of  the  following : 

1.  127  +  (2  +  8)  X  9  +  85  = 

2.  209  —  (27  +  4)  X  5  = 

3.  3300  -f  86  X  6  +  4  = 

4.  3246  —  329  -f  524  x  3  = 

5.  9203  —  6  X  (350  —  239)  = 

6.  (275  +  262)  X  3  —  2  X  (68  —  39)  = 

7.  1935  —  195  +  186  X  4  = 


40  PRACTICAL   ARITHMETIC 

PROBLEMS. 

Let  the  pupil  indicate  the  solution  of  each  problem  by  using  the  sign  X . 

1.  If  sound  moves  1092  feet  in  a  second,  how  far  does  it 
move  in  5  seconds  ? 

Process  Indicated. 
1092  X  5  =  how  far  it  moves. 

Process.  Explanation. 

1092  feet  since  sound  moves  1092  feet  in  1  second,  in  5  seconds  it 

^  moves  5  times  1092  feet ;  1092  feet  X  5  =  5460  feet. 

5460  feet  What  principles  are  involved  ? 

2.  When  wheat  is  worth  $1.25  per  bushel, 
what  is  the  value  of  9  bushels  ? 

3.  What  will  7  horses  cost  at  $175.35  each? 

4.  Since  in  1  mile  there  are  1760  yards,  how  many  yards 
are  there  in  9  miles  ? 

5.  Find  the  cost  of  4886  sheep  at  $6  a  head. 

6.  Each  workman   in  an  iron-foundry  is  paid   $605  a 
year  :  what  do  1 1  men  receive  at  that  rate  ? 

7.  A  bushel  of  corn  weighs  56  pounds :  find  the  weight 
of  12  bushels. 

8.  The  distance  to  the  moon  is  240,000  miles :  what  is  10 
times  that  distance  ? 

9.  If  the  distance  of  the  earth   from  the  sun  is  about 
91,430,000  miles,  how  many  miles  is  9  times  that  distance? 

10.  If  an  army  major  receives  monthly  $208.33,  what  is 
the  monthly  pay  of  1 2  majors  ? 

1 1 .  Mr.  White  owns  3  houses,  and  the  first  house  is  worth 
$3872 ;  the  second,  3  times  as  much ;  and  the  third,  7  times 
as  much.     Find  the  cost  of  the  3  houses. 

Process  Indicated. 

3872  +  (3872  X  3)  +  (3872  X  7)  =  cost  required. 


MULTIPLICATION 


41 


12.  A  farmer's  wife  took  to  a  store  3  pounds  of  butter 
worth  33  cents  a  pound,  and  bought  12  yards  of  calico  at 
$.08  a  yard.     Find  the  balance  due  her. 

13.  Which  are  worth  more,  7  cows  at  $35  apiece  or  3  horses 
at  $75  apiece? 

14.  A  lady  bought  a  bicycle  for  $100 ;  she  rented  it  to  a 
friend  for  5  months  at  $3  a  month,  and  finally  sold  it  for  $75. 
Did  she  gain  or  lose,  and  how  much  ? 

15.  A  man  sold  three  houses  ;  for  the  first  he  received  3575 
dollars ;  for  the  second,  $950  less ;  for  the  third,  three  times 
the  difference  between  the  price  of  the  first  and  second.    What 
did  he  receive  for  the  three  ? 

CHIEF    DIFFICULTY    OF    MULTIPLICATION. 


1.  Multiply  438  by  234. 


1st 
Process. 

2d 
Process. 

438 

438 

234 

234 

1752 

1752 

13140 

1314 

87600 

876 

Explanation. 


IST  PROCESS.- 


f 


102,492    102,492 

Proof. 

234 

438 


1872 
702 
936 

102,492 


4  units          =  4  units. 
234  =  •!  3  tens  =  30  units. 

1 2  hundreds  =  200  units. 

Therefore,  we  are  to  multiply  438  firstly 
by  4  units,  secondly  by  30  units,  thirdly  by 
200  units,  and  then  find  the  sum  of  the 
three  partial  products. 

Multiplying  by  4  we  have  1752  units; 
multiplying  by  30  we  have  13140  units; 
multiplying  by  200  we  have  87600  units. 
The  sum  of  these  products  is  102,492  units. 

2D  PROCESS.— Since  13,140  units  =  1314 
tens,  and  since  87,600  units  =  876  hundreds, 
we  omit  the  ciphers,  and,  writing  1314  as 
tens,  and  876  as  hundreds,  the  significant 
figures  keep  their  relative  positions,  and 
the  result  of  the  addition  is  the  same  as 
before. 


PROOF.— State  the  principle  on  which  the  proof  depends. 


42  PRACTICAL    ARITHMETIC 

2.  Find  the  product,  explain  the  process,  and  show  proof 
of  accuracy : 


(1-) 

(2.) 

(3-) 

(4.) 

(5.) 

317 

483 

847 

657 

343 

15 

16 

23 

26 

37 

(6.) 

(7-) 

(8.) 

(9.) 

(10.) 

$13.35 

$17.85 

$30.23 

$42.93 

$55.66 

33 

43 

72 

81 

29 

Put  a  decimal  point  in  each  product. 

3.  Multiply: 

1.  382  by  19.  11.  619  by  96. 

2.  384  by  35.  12.  777  by  86. 

3.  405  by  53.  13.  910  by  52. 

4.  534  by  37.  14.  732  by  62. 

5.  645  by  73.  15.  839  by  93. 

6.  843  by  54.  16.  327  by  67. 

7.  917  by  46.  17.  931  by  95. 

8.  903  by  74.  18.  733  by  123. 

9.  593  by  91.  19.  981  by  234. 
10.  578  by  83.                          20.  891  by  345. 

4.  How  many  are  837  X  38  ? 

5.  How  many  are  $57.30  X  665  ? 

6.  2537  X  47  =  how  many? 

7.  5386  X  65  =  how  many? 

8.  4860  X  9574  =  ? 

9.  $357.63  X  47  =  ? 

10.  Find  the  second  member  of  the  following  unfinished 
expressions : 

1.  $9.672  X  635  -  3.  51,526  X  527  = 

2.  5732  X  891  =  4.  97,601  X  987  = 


MULTIPLICATION  43 

11.  Multiply  five  thousand  nine  hundred  sixty-five  by  six 
thousand  nine. 

12.  Multiply  four  hundred  sixty -two  thousand  six  hundred 
nine  by  itself. 

13.  Multiply  eight  hundred  forty-nine  million  six  hundred 
seven  thousand  three  hundred  six  by  nine  hundred  thousand 
two  hundred  four. 

14.  Multiply  704  million  130  thousand  496  by  three  thou- 
sand three  hundred  one. 

15.  Multiply  one  hundred  twenty-three  and  45  hundredths 
by  804. 

16.  Multiply  415  and  5  hundredths  by  367. 

17.  Multiply  113  dollars  and  41  cents  by  613. 

18.  Multiply  XLVIII.  by  XIX. 

19.  Multiply  CDLXIV.  by  CDIV. 

20.  Form  an  equation  of  220,056,121,  and  26,626,776. 

21.  Find  the  value  of  (3467  X  7004)  —  (3467  X  704), 
and  form  an  equation. 

PROBLEMS. 

1.  Find  the  cost  of  a  farm  of  202  acres  at  $102  per  acre. 

Process  Indicated. 

$102  X  202  =  cost. 

2.  A  farmer  had  105  rows  of  trees,  each  row  containing  105 
trees.     How  many  trees  had  he  ? 

3.  If  horses  are  worth  $117  each,  and  oxen  $85.50  a  pair, 
what  must  I  pay  for  18  horses  and  5  pairs  of  oxen? 

4.  If  786  yards  of  cloth  can  be  made  in  one  day,  how  many 
yards  can  be  made  in  1252  days? 

5.  A  grocer's  sales  average  $19  a  day  for  the  month  of 
March ;  leaving  out  5  days  for  Sundays,  how  much  money 
did  he  receive  during  the  month  ? 

6.  John  takes  1434  steps  in  going  to  school ;  if  he  goes  and 
returns  twice  a  day,  how  many  steps  will  he  take  in  24  days  ? 


44  PRACTICAL   ARITHMETIC 

7.  There  are  5280  feet  in  a  mile.     How  many  feet  are 
there  in  18  miles? 

8.  If  James  sells  57  papers  a  day  and  Thomas  65  papers, 
how  many  more  does  Thomas  sell  than  James  in  54  days  ? 

Indicate  the  process  by  using  the  signs,  — ,  X  f  =- 

9.  If  I  buy  17  tons  of  iron  at  $38.75  a  ton,  and  26  tons 
at  $40.25  a  ton,  how  much  shall  I  gain  by  selling  the  whole 
at  $42.50  a  ton? 

10.  (?  -[-  ?)  x  ?  —  (?  X  ?  +  ?  X  ?)  =  ?  Substitute  a 
number  for  each  of  the  interrogation  marks  in  the  first  member, 
solve,  and  state  your  problem. 

SHORT   PROCESSES. 

"When  there  are  ciphers  on  the  right  of  multiplicand,  or 
of  multiplier,  or  of  both. 

1.  Multiply  2  by  30. 
Process.  Explanation. 


2 


The  factors  of  30  are  3  and  10.     We  say  "  2  X  8  =  6 ; 


o  rv  and,  by  annexing  a  cipher  to  6,  we  multiply  it  by  10,  and 

have  60." 

60  2.  Multiply  30  by  4. 

3  Q  The  factors  of  30  are  3  and  10.     We  say  "  3  X  4  =  12 ; 

annexing  a  cipher  to  12  multiplies  it  by  10,  and  we  have 
120." 


Does  the  order  in  which   factors  are  used   in 
multiplying  affect  the  result  ? 

40  3.  Multiply  40  by  500. 

5  00  40  =  4  x  10. 

^7^  500  =  5  X  100. 

4  X  5  X  10  X  100  =  20,000. 

We  say  "4  X  5  =  20;  and,  annexing  one  cipher,  we  multiply  by  10 
and  have  200 ;  annexing  two  ciphers  to  that  result,  we  thus  multiply  it  by 
100,  and  have  20,000." 


MULTIPLICATION  45 

BULB. 

Out  off  and  reserve  the  ciphers  on  the  right;  then  mul- 
tiply, and  to  the  product  obtained  annex  the  ciphers 
reserved. 

EXERCISES    AND    PROBLEMS. 

1.  Multiply  486  by  10.     By  100.     By  400. 

2.  Multiply  9560  by  40.     By  80.     By  1000. 

3.  Multiply  2870  by  600.     By  800.     By  900. 

4.  Multiply  2490  by  300.     By  3000.     By  4400. 

5.  Multiply  59,700  by  360.     By  4300.     By  7600. 

6.  Multiply  42,300  by  320.     By  3700.     By  57,000. 

7.  Multiply  4,871,000  by  270,000.     By  304,000. 

8.  Multiply  $7849.93  X  400.     By  5000. 

9.  Multiply  600,700  X  6000.     By  4,004,000. 
10.  Multiply  CDXL.  by  M.     By  LIX. 

Process  H*  ^  tne  7earty  Pav  °^  a  rear-admiral  is 

$6000,  how  much  will  he  have  received  in  20 

0         y^t 

12.  One    mile   contains    5280    feet.      How 


$120,OOC  many  feet  in  60Q  mileg  ? 

13.  There  are  350  rows  of  trees  in  an  orchard,  120  trees  in 
a  row,  and  3000  apples  on  each  tree.     How  many  apples  in 
the  orchard  ? 

14.  One  acre  contains  160  square  rods.     How  many  square 
rods  in  300  acres  ? 

15.  One   pound   avoirdupois   contains    7000   troy   grains. 
How  many  grains  in  230  pounds  ? 

16.  A  short  ton  ==  2000  pounds.     How  many  pounds  in 
570  tons? 

17.  The  circumference  of  the  earth  =  about  25,000  miles. 
One   mile  =  1760   yards.      How  many  yards   around   the 
earth? 


46  PRACTICAL  ARITHMETIC 

18.  If  one  bushel  of  corn  costs  $.65,  what  will  1000  bushels 
cost? 

19.  At  $160  an  acre,  what  will  500  acres  cost? 

20.  One  hour  =  60   minutes ;   one  minute  =  60  seconds. 
How  many  seconds  in  24  hours  ? 


MULTIPLICATION   BY   FACTORS. 

1.  You  have  learned  that  the  multiplicand  and  multiplier 
are  called  the  factors  of  the  product. 

What  factors  will  produce  4  ?     6?     8?     10?     12?     15? 
16?     18? 

2.  All  numbers  that  can  be  thus  factored  are  called  Com- 
posite numbers. 

PRINCIPLE. 

Multiplication  may  be  performed  by  using  the  factors 
of  the  multiplier. 

EXERCISES. 
1.  Multiply  5  by  6,  using  the  factors  of  6. 

Process.  Explanation. 

6=3x2.  6  =  3x2;  therefore,  we  say  "  6  times  5  =  2  times  3 

times  5 ;  3  times  5  =  15,  and  2  times  15  =  30." 


-^,30 


5 

Proof. 

3 

•X- 

-x- 
•X- 

•X-  -X-  -X- 
•X-  -X-  -X- 
•X-  -X-  -X- 

*   ) 

•x-     >•  —  1  0 

*  ) 

15 

2 

•x- 

•x-  *  * 

*    1 

•x-   -x-   -x- 


PROOF. — Six  rows  of  5  stars  each  =  5X6  =30,  the  whole  number.  3 
rows  of  5  stars  each  =  5x3  =  15.  2  groups  of  15  each  =  16  X  2  = 
30,  the  whole  number. 


MULTIPLICATION  47 

2.  Multiply  438  by  15. 
Process.  Explanation. 

5  __  5  x  3.  Since  the  factors  of  15  are  5  and  3,  we  first,  for 

convenience,  multiply  by  the  larger  factor,  5,  and 
438  that  result  by  3,  and  obtain  6570. 

5 

Proof. 

2190  438X15  =  6570. 


6570  3.  Multiply,  using  factors  : 

1.  6809  by  49.         5.  91,849  by  36. 

2.  435,261  by  63.       6.  4953  by  81. 

3.  310,204  by  48.       7.  14,953  by  144. 

4.  97,387  by  45.        8.  2348  by  21. 

PROBLEMS   COMBINING-   ADDITION,   SUBTRACTION, 

AND   MULTIPLICATION. 

First  indicate  the  process. 

1.  I  have  10  bags  of  coffee,  each  containing  50  pounds. 
How  many  pounds  of  coffee  have  I  ? 

2.  If  hay  is  worth  $14.50  per  ton,  and  oats  $.56  a  bushel, 
what  will  be  the  cost  of  27  tons  of  hay  and  200  bushels  of 
oats? 

3.  A  drover  bought  43  cows  at  $22  each,  64  sheep  at  $13 
each,  and  16  horses  at  $135  each,  and  sold  them  all  for  $4010. 
How  much  did  he  gain  ? 

4.  A  freight  train  consists  of  28  cars,  and  each  car  contains 
136  casks  of  lime,  weighing  240  pounds  each.     How  many 
pounds  of  lime  in  the  whole  cargo  ? 

5.  Sound  travels  at  the  rate  of  1092  feet  in  a  second.     If 
between  the  flash  of  lightning  and  the  clap  of  thunder  there 
were  9  seconds,  how  far  distant  was  the  cloud  that  produced 
the  flash  ? 


48  PKACTICAL  ARITHMETIC 

6.  If  250  pounds  of  charcoal  are  used  in  making  a  ton 
of  gunpowder,  how  many  pounds  will  be  used  for  1280  tons 
of  gunpowder? 

7.  What  sum  of  money  will  be  required  to  pay  a  regi- 
ment of  987  men  for  a  year's  services,  at  $18  a  month  for 
each  man  ? 

8.  A  merchant  sold   324  barrels  of  apples  at  $4.75  a 
barrel,  and  gained  $162  on  the  transaction.     What  did  his 
apples  cost  him  ? 

9.  A  cattle  train  is  made  up  of  17  cars,  and  each  car  con- 
tains 53  sheep.     The  average  weight  of  the  sheep  is  115 
pounds.     How  much  do  they  all  weigh  ? 

10.  A  farmer  bought  a  farm  containing  10  fields;  3  of  the 
fields  contained  9  acres  each;  3  other  of  the  fields,  12  acres 
each  ;  the  remaining  4  fields,  each  1 5  acres.    How  many  acres 
in  the  farm  ?    What  was  the  cost  of  the  farm  at  $21  an  acre  ? 

11.  If  it  requires  1716  pickets  to  fence  one  side  of  a  square 
lot,  how  many  pickets  will  be  required  to  fence  13  lots  of  the 
same  size  and  shape  ? 

12.  Mrs.  Brown  bought  12  yards  of  oilcloth  at  65  cents 
a  yard,  and  32  yards  of  ingrain  carpet  at  75  cents  a  yard. 
What  did  she  pay  for  all  ? 

13.  One  day  =  86,400  seconds ;  one  year  =  365  days.     If 
light  moves  at  the  rate  of  186,000  miles  in  a  second,  how  far 
distant  is  a  star  whose  light  is  one  year  in  reaching  the  earth  ? 

14.  An  army  lost  in  battle  315  killed,  417  wounded;  the 
enemy  lost  in  killed  and  wounded  17  times  as  many.     How 
many  soldiers  were  killed  and  wounded  in  this  battle? 

1 5.  If  1327  barrels  of  flour  will  feed  the  inhabitants  of  a  city 
for  one  day,  how  many  barrels  will  supply  them  for  two  years  ? 

16.  The  Erie  Railroad  is  about  425  miles  long,  and  cost 
sixty-five  thousand  dollars  a  mile.     When  9,645,635  dollars 
were  paid,  what  was  the  balance  due  ? 


MULTIPLICATION  49 

17.  Two  vessels  are  4500  miles  apart,  and  travel  toward 
each  other;  one  at  the  rate  of  91  miles  a  day,  and  the  other 
at  the  rate  of  85  miles  a  day.     How  far  apart  are  they  at  the 
end  of  24  days? 

18.  Two  vessels  start  from  New  York  for  Liverpool;  one 
sails  at  the  rate  of  138  miles  a  day ;  the  other,  at  the  rate  of 
215  miles  a  day.     How  far  will  they  be  apart  at  the  end  of 
nine  days? 

19.  What  is  the  product  of  three  hundred  eleven  million 
two  hundred  twenty-one  thousand  multiplied  by  two  hundred 
three  thousand  one  hundred  five  ? 

MISCELLANEOUS   EXERCISES. 

1.  Complete  these  equations:    (16  —  11+2)X5  =  ? 
(4  +  15)  X  (15  —  4)  X  6  =  ? 

2.  Use  the  signs,  (),+,—,  X,  =,  in  forming  an  equation 
of  your  own. 

3.  An  unfinished  equation  is :  63,915  +  (?)  =  one  million. 
Find  the  required  part. 

4.  (?)  +  4872  =  8021.     Complete  the  equation. 

5.  5301  —  (?)  =  4255.     Complete  the  equation. 

6.  Multiply  three  million  three  by  one  hundred  thousand 
one. 

7.  Write  the  immediately  preceding  numbers  in  Roman 
numerals. 

8.  Multiply  3008  by  132,  using  the  two  factors  of  132 
whose  difference  is  1. 

9.  Find  the  value  of  6145  —  3408  +  1931  X  3400  — 
(33,600  X  105). 

10.  A  drover  had  690  sheep ;  he  sold  340  to  one  man,  324 
to  another,  and  then  bought  enough  to  make  his  number  700. 
How  many  did  he  buy  ? 


50  PRACTICAL    ARITHMETIC 

REVIEW. 

1.  Define  the  following  terms  : 

1.  Multiplication.  7.  Parenthesis. 

2.  Multiplicand.  8.  Yinculum. 

3.  Multiplier.  9.  Proof. 

4.  Product.  10.  Members. 

5.  Factors.  11.  Composite  number. 

6.  Sign  of  Multiplication.    12.  Equation. 

2.  Repeat  the  principles  of  Multiplication. 

3.  Repeat  the  rule  for  Multiplication  when  the  factors  are 
composed  of  significant  figures  with  ciphers  on  their  right. 

4.  What  is  the  principle  respecting  factors  of  the  multiplier  ? 

5.  Illustrate  the  principle. 

6.  Invent  five  problems  that  will  involve  Addition,  Sub- 
traction, and   Multiplication.     Indicate  the  solution   by  the 
signs,  +,  — ,  X.  

DIVISION. 

INDUCTIVE    STEPS. 

1.  Since  2X3  =  6,  the  2  and  3  are  called  what? 

2.  Then  if  2  is  one  factor  of  6,  what  is  the  other  ? 

3.  Why  must  3  be  the  other  ? 

4.  Because  2  times  3  is  6,  we  say  that  2  is  contained  in  6 
three  times. 

5.  How  many  times  is  4  contained  in  8  ? 
Process.  Explanation. 

4)8(2  We  say  "  4  is  contained  in  8  two  times,  because  2  times 

s_        4  =  8-" 

0  6.  How  many  times  is  5  contained  in  10? 

Write  the  process  and  explain. 

7.  The  process  of  finding  how  many  times  one  number  is 
contained  in  another  is  called  Dividing. 


DIVISION  51 

8.  Divide  16   by  8,  and   show  that  division   is  a  short 
method  of  subtraction. 

1st  Process.         2d  Process.  Explanation. 

16  \  8)16(2  IST    PROCESS —We    say    "  Sub- 

j-  Subtract.  tracting  8  once,  we  have  8  remaining ; 

subtracting  8  a  second  time,  we  have 
Subtract  0  remaining;  therefore,  8  is  contained 


8   '  in  16  two  times. 

0  2o  PROCESS. — Since  8  times  2  = 

16,  8  is  contained  in  16  two  times. 

9.  One  factor  of  24  is  3,  what  is  the  other  factor? 

10.  Dividing  by  7  separates  a  number  into  how  many  equal 
parts  ? 

One  of  seven  equal  parts  is  called  one-seventh,  written  \. 
|  of  28  equals  what  ? 

11.  If  a  farmer  pays  $28  for  7  sheep,  how  much  is  that 
apiece  ? 

Process.  Explanation. 

7  )  28  (  4  Since  he  pays  $28  for  7  sheep,  he  pays  for  each  one- 

2g  seventh  of  $28,  or  $4. 

0  Did  you  divide  by  7  sheep,  or  simply  by  7  ? 

Is  7,  then,  an  abstract  or  a  concrete  number  ? 
Is  your  answer  4  or  $4  ? 

12.  How  many  sheep  at  $4  apiece  can  a  farmer  buy  for 
$28? 

if  $4  is  one  factor  of  $28,  is  $7  the  other  factor  ?  Is  7 
sheep  ?  Is  7  ? 

What,  however,  does  the  7  indicate? 

DEFINITIONS. 

1.  Division  is  the  process  of  finding  how  many  times  one 
number  is  contained  in  another,  or  of  finding  one  of  the  equal 
parts  of  a  number. 

NOTE. — This  latter  operation  is  called  Partition. 


52  PEACTICAL    AEITHMETIC 

2.  The  Dividend  is  the  number  to  be  divided. 

3.  The  Divisor  is  the  number  by  which  we  divide. 

4.  The  Quotient  is  the  result  obtained. 

5.  The  number  which  is  sometimes  left  after  dividing  is 
called  the  Remainder. 

When  the  remainder  is  0,  the  division  is  said  to  be  exact. 

6.  The  Sign  of  Division  is  -*-, 

21  H-  7  =  3,  is  read  "  21  divided  by  7  equals  3." 

Take  notice  that  the  dividend  is  written  before  the  sign  ;   the  divisor 
after  the  sign. 

In  practice  it  is  found  convenient  to  indicate  division  thus  : 

7)21  7)21(3  21 

—     or  thus  :  v        or  thus  :  —  =  3. 


PRINCIPLES. 

1.  Dividing  a  number  by  one  of   its  factors  gives  the 
other  factor  for  the  quotient. 

2.  When  the  divisor  is  an  abstract  number,  the  divi- 
dend and  quotient  are  like  numbers. 

3.  When  the  dividend  and  divisor  are  like  numbers,  the 
quotient  is  an  abstract  number. 

4.  The  divisor  multiplied  by  the  quotient  reproduces  the 
dividend. 

EXERCISES 

FOR  ANALYTIC  AND  SYNTHETIC  EXPLANATION. 
The  Divisor  not  exceeding  12. 

1.  If  6  is  one  factor  of  24,  what  is  the  other? 

Process.  Explanation. 

Divisor.  Dividend.  Quotient.  "\ye  say  u  gince  6  is  one  factor  of  24,  and 

6  )     24     (4  since  6  times  4  =  24,  4  is  the  quotient,  or 

24  other  factor.  '  ' 
/-v  State  the  principle. 


DIVISION  53 

2.  474  has  a  factor,  2 ;  find  the  other  factor. 
Process.  Explanation. 

2)474(237  We  say  "474  =  4  hundreds,  7  tens,  4  units.     4 

4  hundreds  -=-2  =  2  hundreds.     Bring  down  7  tens  ;  7 

tens  -=-2  =  3  tens  and  1  ten  remaining.     1  ten  and  4 

'  units  =  14  units  ;  14  units  -=-2  =  7  units.     The  quo- 

6  tient  is  2  hundreds,  3  tens,  7  units,  or  237. 

14  Proof. 

14  i  237  X  2  =  474. 

0  State  the  principle. 

3.  5  is  one  factor  of  35  ;  find  the  other. 

4.  8  is  one  factor  of  48  ;  find  the  other. 

5.  7  is  one  factor  of  49  ;  find  the  other. 

6.  9  is  one  factor  of  72  ;  find  the  other. 

7.  12  is  one  factor  of  108 ;  find  the  other. 

8.  11  is  one  factor  of  132  ;  find  the  other. 

9.  12  is  one  factor  of  144  ;  find  the  other. 
10.  Divide  144  by  8. 

We  may  express  the  division  in  four  different  ways : 
(1.)  (2.)  (3.) 

8)144(18  8)144  144 

8_  ~18~  ~8~ 

64 

M  (4.) 

0  144  -5-  8  ==  18 

The  first  is  called  Long  Division ;  the  second,  third,  and 
fourth,  Short  Division. 

Explanation. 

144  =  1  hundred  4  tens  4  units.  1  is  not  divisible  by  8,  hence  we  say 
"  1  hundred  -(-  4  tens  =  14  tens  ;  14  tens  -?-  8  =  1  ten,  with  6  tens  remain- 
ing ;  6  tens  -f-  4  units  =  64  units ;  64  units  -=-8  =  8  units,  with  0  units 
remaining.  Therefore  18  is  the  exact  quotient." 

Proof. 

18  X  8  =  144 


54 


PRACTICAL   ARITHMETIC 


11.  Solve  by  Long  Division,  explain,  and  prove  the  fol- 
lowing : 

I1-)  (2.)  (3.)  (4.) 


3)849( 


5)940( 


7)497 


8)992( 


Process. 

5)7506(1501 
5_ 
25 
25 


06 
_5^ 
1   rem. 


(5.) 


(6.) 


(7-) 


9)  7506  (         10)  41,690  (         11)  103,961  ( 

(8.)  (9-)  (10.) 

12)  113,820  (   11)  57,893  (   12)  74,856  ( 


(11.) 
8 )  38,496  ( 


(12.) 

9)  43,281  ( 


(13.) 
12) 2964 ( 


(14.) 

3)  12,414  ( 


(15.) 
5 )  32,795  ( 


(16.) 

4)  374,864  ( 


(17.) 
3)629,274 


(18.) 
5)  947,860  ( 


(19.) 

6)$1589.10( 


(20.) 
7)  $6472.69  ( 


(21.) 

8)  $1025.68  ( 


(22.) 

9)  $1999.98  ( 


Place  a  decimal  point  in  the  quotient. 

12.  Solve  by  Short  Division  the  following: 

(1.)  (2.)  (3.)  (4.) 

6)1698     10)1980     7)994      8)1984 


(5.)  (6.)  (7.)  (8.) 

9)15,012        10)41,690     11 )  103,961     12)113,820 


DIVISION  55 

(9.)  (10.)  (11.)  (12.) 

11)57,893      12)74,856       8 )  38,496         9)43,281 

(13.)  (14.)  (15.)  (16.) 

7)193,760      12)29,640       3 )  24,828        5  )  32,795 

(17.)  (18.)  (19.)  (20.) 

4)374,864     3)629,274      5)947,860      6)158,910 

(21.)  (22.)  (23.) 

$48.56     _9  $6.44         .  $976.50  = 

879 

Place  a  decimal  point  in  the  quotient. 

(24.)  (25.)  (26.) 

Pounds. 

t-  {y  Q  Bushels.  Rods. 

'    -      =  ?     94,000  -H-  8  =  ?    760,344  -=-  12  =  ? 


In  (24),  (25),  (26),  are  your  quotients  abstract  or  concrete? 
Give  the  principle. 

The  Divisor  exceeding  12,  but  less  than  1OO. 
1.  Divide  34,028  by  13. 

Process.  Explanation. 

th.  h.t.u.  th.h.t.u.  34,028  =  34  thousands  0  hundreds  2  tens  8 

13)34,028(2617  units.     34  thousands  -=-  13  =  2  thousands,  with 

26  8  thousands  remaining ;  8  thousands  -f-  0  hun- 

~8Q~h.  dreds  =  80  hundreds ;  80  hundreds  -=-13  =  6 

7  hundreds,  with  2  hundreds  remaining;  2  hun- 
dreds -f  2  tens  ==  22  tens ;  22  tens  -=-13  =  1 

22   '  ten,  with  9  tens  remaining  ;  9  tens  -f-  8  units  — 

1 3  98  units  ;  98  units  -=-13  =  7  units,  with  7  units 

go  u.  remaining. 

91  2.  Divide  5684  by  14. 

7  3.  Divide  6480  by  15. 


56  PRACTICAL   ARITHMETIC 

4.  Divide  2672  by  16.  13.  1504  —  47  =  ? 

5.  Divide  2928  by  17.  14.  5289  —  43  =  ? 

6.  Divide  4147  by  18.  15.  9464  —  52  =  ? 

7.  Divide  8797  by  19.  16.  5612  —  61  —  ? 

8.  Divide  6872  by  24.  17.  6336  —  72  =  ? 

9.  Divide  8519  by  27.  18.  4557  —  21  =  ? 

10.  Divide  9672  by  31.  19.  3264  —  24  =  ? 

11.  Divide  28,490  by  15.  20.  2664  —  37  =  ? 

12.  Divide  18,476  by  42.  21.  3465  —  99  =  ? 

PROBLEMS. 

1.  3  feet  =  1  yard.     How  many  yards  in  927  feet? 

Process.  Explanation. 

LONG  DIVISION.  Since  3  feet  =  1  yard,  927  feet  equal  as  many 

feet.    feet.  yards  as  3  is  contained  times  in  927.     927  -e-  3  = 

3  )  927  (  309  309.     Therefore  927  feet  equal  309  yards. 

_ How  is  the  0  in  the  quotient  obtained  ? 

In  the  process,  is  309  an  abstract  or  a 
^'  concrete  number ?     State  the  principle. 

Is  the  process  long  or  short  division  ? 
Write  the  process  in  short  division  and  explain  the  steps. 

2.  I  paid  285  cents  for  a  railroad  ticket  at  3  cents  a  mile. 
How  many  miles  did  I  ride  ? 

3.  If  you  buy  12  pounds  of  soap  for  96  cents,  how  much 
do  you  pay  for  a  pound  ? 

4.  If  the  circumference  of  a  wheel  is  12  inches,  how  many 
times  will  it  revolve  in  moving  1728  inches  ? 

5.  If  it  takes  5  bushels  of  wheat  to  make  a  barrel  of  flour, 
how  many  barrels  can  be  made  from  65,890  bushels? 

6.  A  merchant  has  1620  yards  of  calico,  which  he  wishes 
to  cut  into  15-yard  patterns.     How  many  patterns  will  he 
have? 


DIVISION  57 

7.  How  many  times  must  you  take  7  dollars  to  make  567 
dollars  ? 

8.  A  boat  sails  1872  miles,  going  at  the  rate  of  18  miles 
an  hour.     How  many  hours  does  it  sail  ? 

9.  How  many  sacks,  each  containing  55  pounds,  can  be 
filled  from  2035  pounds  of  flour? 

10.  If  I  divided  $570  equally  among  some  men,  gi\'ng  to 
each  man  $8.00,  how  many  men  were  there? 

11.  If  4  weeks  make  a  month,  how  many  months  are  there 
in  264  weeks  ? 

12.  Into  how  many  lots  of  39  acres  each  can  a  tract  of 
land  containing  6318  acres  be  divided? 

13.  The  circumference  of  a  wheel  of  a  bicycle  is  7  feet. 
How  many  revolutions  will  it  make  in  going  18,480  feet? 

14.  How  many  sheep  at  $9  a  head  can    be  bought  for 
$1377? 

15.  A  man  bought  land  at  $87  an  acre,  paying  $31,755  for 
it.     How  many  acres  did  he  buy? 

16.  How  many  feet  are  there  in  a  mile,  if  42  miles  contain 
22 1,760  feet? 

17.  Divide  four  hundred  eighteen   thousand  six  hundred 
forty-eight  by  twenty-four. 

18.  If  the  post-office  sends  13,125  pounds  of  mail-matter 
in  bags,  each  holding   75   pounds,  how  many   bags  will  it 
require  ? 

19.  An  estate  worth  2943  dollars  is  to  be  divided  equally 
among  a  father,  mother,  3  daughters,  and  4  sons.     What  will 
be  the  portion  of  each  ? 

20.  Solve  the  following  equation  : 

1  798 
(5280  —  1760  -f  144)  x  -^   -*-  12T=  ? 

i  — 

21.  I  bought  15  horses  at  $75  a  head;  at  how  much  per 
head  must  I  sell  them  to  gain  $210? 


58 


PRACTICAL    ARITHMETIC 


EXERCISES. 

The  Divisor  exceeding-  1OO. 
1.  Divide  145,260  by  108. 


Process. 

108)145,260(1345 
108 
372 
324 


486 

432 
540 
540 


Explanation. 

145  thousand  -f-  108  ==  1  thousand,  with 
37  thousand  remaining;  37  thousand  -f-  2 
hundred  =  372  hundred;  372  hundred  -f- 
108  =  3  hundred,  with  48  hundred  remain- 
ing ;  48  hundred  -f  6  tens  =  486  tens ;  486 
tens  ~  108  —  4  tens,  with  54  tens  remain- 
ing;  54  tens  -j-  0  units  =  540  units;  540 
units  -f-  108  =  5,  with  0  remaining. 


1.  1,874,774  by  162. 

2.  1,206,528  by  192. 

3.  815,898  by  421. 

4.  199,864  by  301. 

5.  315,008  by  428. 
3.  Find  the  value  of: 

1.  395,630  —  750. 

2.  683,537  —  987. 

3.  900,503  —  173. 

4.  456,007  —  560. 

5.  881,881  —  700. 

6.  341,517  —  529. 

7.  237,607  —  837. 


2.  Divide: 

6.  503,652  by  564. 

7.  705,776  by  728. 

8.  892,696  by  839. 

9.  902,260  by  916. 
10.  683,537  by  987. 


8.  1,056,566  -f-  326. 

9.  10,365,051  -r-3021. 

10.  2,159,450  -T-  2465. 

11.  496,839,715  —  1047. 

12.  9,325,814  ~  2042. 

13.  27,227,704 -f-  6472. 

14.  47,254,149  -f-  4674. 


Find  the  value  of: 

1.  352,107,193,214-4-210,472. 

2.  558,001,172,606,176,724  -f-  2,708,630,425. 

3.  1 23,456,789,102,345,678 -r-  1,234,567,890. 

4.  987,654,321,000,000,000 -f-  9,876,543,210. 

5.  2,016,722,783,975,663,729  -4-  41,927,081. 


DIVISION  59 


PROBLEMS. 


1.  A  man  has  12,000  dollars  to  invest  in  land.      How 
many  acres  can  he  buy  at  125  dollars  an  acre? 

2.  There  are  47,520  yards  in  27  miles.    How  many  yards 
are  there  in  one  mile? 

3.  There  are  640   acres   in  a  square  mile.     How  many 
square  miles  are  there  in  the  District  of  Columbia,  which  con- 
tains 38,400  acres  ? 

4.  If  the  earth  in  its  revolution   round   the  sun  moves 
1,641,600  miles  a  day,  how  far  does  it  move  in  one  second,  a 
day  containing  86,400  seconds? 

5.  If  4671  building  lots  are  worth  1,985,175  dollars,  how 
much  is  one  building  lot  worth? 

6.  What  number  of  dollars  must  be  multiplied  by  124  to 
produce  40,796  dollars  ? 

7.  There  are  31,173  verses  in  the   Bible.      How   many 
verses  must  be  read  each  day,  that  it  may  be  rend  through  in 
a  common  year  ? 

8.  Pennsylvania  contains  45,125  square  miles,  and  Dela- 
ware contains  2050  square  miles.     How  many  states  the  size 
of  Delaware  could  be  made  from  Pennsylvania? 

9.  How  long  can  125  men  subsist  on  an  amount  of  food 
that  will  last  one  man  4500  days? 

10.  If  1988  hogsheads  of  molasses  cost  115,304  dollars, 
what  will  one  hogshead  cost? 

11.  A  balloon  is  said  to  have  ascended  37,000  feet.     How 
many  miles  ?     (One  mile  =  5280  feet.) 

12.  If  one  of  two  factors  of  4,312,695  is  1205,  what  is  the 
other  factor  ? 

13.  A  man  has  8000  dollars ;  he  buys  two  houses  for  4500 
dollars,  and  invests  the  remainder  in  land  at  140  dollars  an 
acre.     How  many  acres  of  land  can  he  buy  ? 


60  PRACTICAL   ARITHMETIC 

14.  If  the  distance  from  the  earth  to  the  sun  is  91,430,000 
miles,  how  long  will  it  take  light  from  the  sun  to  reach  us, 
if  it  moves  186,000  miles  a  second? 

15.  How  many  years  will  it  take  a  man  to  save  $5475,  if 
his  savings  average  one  dollar  per  day,  reckoning  365  days  to 
the  year  ? 

16.  A  railroad  that  cost  $4,076,500  was  divided  into  8153 
equal  shares.     What  was  the  cost  of  each  share  ? 

17.  There  are  231  cubic  inches  in  a  gallon.     How  many 
gallons  in  a  tank  that  contains  139,755  cubic  inches? 

18.  The  salary  of  the  President  of  the  United   States  is 
$50,000  a  year.     How  much  does  he  receive  each  day  ? 

19.  If  a  pound  of  cotton  can  be  spun  into  a  thread  70  miles 
long,  how  many  pounds  of  it  must  be  spun  to  reach  around 
the  world,  a  distance  of  25,000  miles  ? 

20.  Two  trains  on  the  same  railway  are  689  miles  apart. 
If  they  start  at  the  same  time  and  run  toward  each  other,  one 
averaging  27  miles  per  hour,  and  the  other  26  miles,  in  how 
many  hours  will  they  meet? 

21.  Find  the  value  of  15  X  37,153  —  73,474  —  67,152 
-s-  4  -f  40,734  X  2. 

Suggestion  :  Use  X  and  -=--  first. 

22.  Find  the  value  of  (7854  —  4913)  X  3  —  (20,352  — 
5194)  _|_  53  _  6  -f  (395,456  —  2364)  -*-  556. 

23.  Find  the  value  of  (12  -f  7  —  9)  X  5  -f-  10. 

24.  Find  the  value  of  (5  +  7  —  3)  X  3  +  (3  -f  5  —  4) 
--4. 

25.  Find  the  value  of  (828  —  475  —  325)  +  (982  —  620 
—  82). 

26.  Find  the  value  of  849  X  4  ~  3  —  714  X  4  -5-  3  — 
135  X  4  -=-  3. 

27.  Find  the  value  of  (LI.  —  III.  +  I.)  -5-  VII.  +  (III. 
X  V.  —  IX.)  -5r  III. 


DIVISION  61 

28.  Find  the  value  of  (XXVII.  +  XXII.  —  XIX)  x  VI. 

29.  Find  the  value  of  (CCCLXI.  —  CGI.)  X  (CCCXX. 
—  CCCXIL). 

30.  Find  a  second  factor  of  4807,  taking  11  as  the  first 
factor. 

MISCELLANEOUS   EXERCISES. 

1.  The  minuend  is  900,000  and  the  subtrahend  is  323,456. 
What  is  the  difference  ? 

2.  The  minuend  is  300,400  and   the  difference  197,325. 
Find  the  subtrahend. 

3.  The  subtrahend  is  204,054  and  the  difference  is  9735. 
What  is  the  minuend? 

4.  The  product  of  two  numbers  is  567,204,  and  one  of  the 
numbers  is  141,801.     Find  the  other  number. 

5.  The  multiplier  is  3007  and  the  multiplicand  is  3007. 
What  is  the  product  ? 

6.  The  product  is  24,483  and  the  multiplier  is  3.     What 
is  the  multiplicand? 

7.  The  product  is  24,402  and  the  multiplier  is  21.     Find 
the  multiplicand  ? 

8.  The  product  is  20,692  and  the  multiplicand  is  739. 
Find  the  multiplier. 

9.  The  divisor  is  437,  the  quotient  is  730,  and  the  re- 
mainder is  89.     What  is  the  dividend? 

10.  The  divisor  is  954,  the  quotient  is  840,  the  remainder 
227.     Find  the  dividend. 

11.  What  number  divided  by  573  will  give  a  quotient  of 
205  and  a  remainder  of  89  ? 

12.  Of  what  number  is  623  both  the  divisor  and  the  quo- 
tient? 

13.  The  sum  of  two  numbers  is  21,000,000 ;  one  of  the 
numbers  is  12,113,141.     Find  the  other  number. 

14.  Divide  18,490,700  by  73,000. 


62  PRACTICAL   ARITHMETIC 

15.  Multiply  5690  by  3008.     Prove  by  division. 

16.  Show  that  (26  X  26  —  15  X  15)  -5-  (26  +  15)  =  26 
-  15. 

17.  How  many  times  in  succession  can  3589  be  subtracted 
from  241,462  ?     What  will  be  the  remainder  ? 

18.  A  certain  number  is  contained  41  times  in  1043,  with 
1 8  as  a  remainder.     What  is  the  number  ? 

19.  What  number  is  that  which,  divided  by  12,  the  quo- 
tient multiplied  by  8,  and  580  added  to  the  product,  equals 
740? 

20.  Divide  9,999,999  by  33,300. 

MISCELLANEOUS   PROBLEMS. 

1.  If  a  ship  sails  10  miles  an  hour,  in  how  many  days  will 
it  cross  the  Atlantic  Ocean,  2880  miles? 

Process  Indicated. 

2880  -+-  10 


24 


=  the  number  of  days  required. 


Process.  Explanation. 

2880      _  ~        .  1.  Since  the  ship  sails  10  miles  an  hour, 

10  it  will  sail  2^80  miles  in  2880  -=-  10  =  288 


24)  288  (12  days. 

2.  Since  24  hours  ==  1  day,  288  hours  = 


288  , 
48  24"  ****  =  12  da-vs> 

NOTE.—  Carefully  indicate  each  solution. 

2.  How  many  barrels  of  apples,  at  $2.75  a  barrel,  must  be 
given  for  6  barrels  of  cranberries,  at  $8.25  a  barrel  ? 

3.  How  many  pounds  of  coffee,  worth  $.12  a  pound,  must 
be  given  for  368  pounds  of  sugar,  worth  $.09  a  pound  ? 

4.  A  young  farmer  earns  $60  a  month  and  spends  $25. 
In  what  time  can  he  save  enough  to  pay  for  a  farm  of  50 
acres,  at  $28  an  acre? 


DIVISION  63 

5.  A  grocer  bought  250  pounds  of  coffee  for  $82.50,  and 
sold  it  at  $.37  a  pound.     What  did  he  gain? 

6.  (309  —  76)  +  (4426  —  309)  +  (6375  —  4426)  -f  76 
=  9375  is  a  defective  equation  to  what  extent? 

7.  There  were  24,012  public  schools  in  Pennsylvania  in 
1893,  with  994,407  pupils.    How  many  pupils,  on  an  average, 
in  each  school  ? 

8.  Multiply  the  sum  of  276  and  347  by  three  times  their 
difference  ? 

9.  A  park  is  48  rods  long  and  32  rods  wide.    How  many 
times  must  a  boy  go  around  it  on  his  bicycle  to  travel  45  miles, 
there  being  320  rods  in  a  mile  ?     How  many  times  must  he 
go  around  the  park  to  travel  one  mile? 

10.  A  man  dyiug,  left  three  tracts  of  land  to  be  divided 
equally  among  his  six  children.    The  first  tract  contained  1118 
acres;  the  second,  three  times  as  much  lacking  193  acres;  the 
third,  twice  as  much  as  the  other  two  lacking  105  acres.    What 
was  each  one's  share? 

11.  A/s  house  cost  $7825,  which  was  $4218  less  than  the 
cost  of  the  farm.     What  was  the  cost  of  both  ? 

12.  The  diameter  of  the  earth  at  the  poles  is  41,707,620 
feet,  and  at  the  equator,  41,847,426  feet.     How  much  does 
the  equatorial  diameter  exceed  the  polar  diameter? 

13.  What  will  53,000  bricks  cost  at  $7.25  per  M.? 

14.  Mr.  Gill,  a  drover,  purchased  36  head  of  cattle,  at  $64 
a  head,  and  88  sheep,  at  $5.00  a  head.     He  sold  the  cattle  for 
$40.00  a  head,  and  the  sheep  for  $4.00  apiece^    Did  he  lose, 
and  how  much  ? 

15.  Of  two  boys,  one  was  lazy  and  did  not  rise  till  nine 
o'clock,  while  the  other  was  active  and  rose  every  morning  at 
six.    Allowing  365  days  to  the  year,  how  many  hours  did  the 
lazy  boy  lose  in  five  years  ? 

16.  There  are  two  numbers,  the  greater  of  which  is   25 


64  PRACTICAL    ARITHMETIC 

times  670,  and  their  difference  55  times  81.     Find  the  less 
number. 

17.  I  bought  87  acres  of  land  at  $50  an  acre,  and  paid 
$3150  in  cash,  and  the  balance  in  labor  at  $240  a  year.     How 
many  years  of  labor  did  it  take  ? 

18.  A  farmer  has  1000  head  of  cattle  in  five  fields.    In  the 
first  he  has  315  head;  in  the  second,  175  head ;  in  the  third, 
300  head ;  and  in  the  fourth,  the  same  number  as  in  the  fifth. 
How  many  has  he  in  the  fifth  ? 

19.  If  a  man  sells  19  bushels  of  potatoes  at  $.55  a  bushel, 
23  bushels  of  oats  at  $.53  a  bushel,  and  with  the  proceeds 
buys  8  yards  of  broadcloth,  how  much  does  he  pay  a  yard  for 
the  broadcloth  ? 

20.  If  a  newsboy  buys  papers  at  $.08  a  dozen,  and  sells 
them  at  $.01  apiece,  how  much  can  he  clear  in  March,  if  he 
averages  1 20  papers  a  day  ? 

Suggestion:   (.01  X  12  —  .08)  X  ~j  X  31  =  ? 


ANALYSIS   AND   REVIEW. 

"Analysis  reasons  from  the  given  number  to  one,  and 
from  one  to  the  required  number." 

1.  A  man  bought  13  horses  for  $2405.     What  would  he 
pay  for  37  horses  at  the  same  rate  ? 

Process  Indicated.    * 

$2405  X  37  =  BUmreceived. 

13 
Process.  Explanation. 

13  horses  cost  $2405.  *•  since  13  horses  cost  $2405>  l  horse 

-,   i  *-,  uf-  will  cost  $2405  -=-  13,  or  $185. 

1  horse  costs  $185. 


g?  horses  win 
37  horses  cost  $6845.         cost  $185  x  87  or  $6845. 


DIVISION  65 

Let  the  pupil  indicate  the  solution  hy  using  the  appropriate  signs. 

2.  If  25  pounds  of  sugar  cost  $2.50,  what  will  36  pounds 
cost? 

3.  If  I  exchanged   40  barrels  of  flour  for  61   yards  of 
cloth  at  $4  a  yard,  how  much  did  I  get  per  barrel  for  the 

flour? 

Indicate  the  process  and  explain. 

4.  A  carriage  maker  sold  15  carriages  for  $1875.     How 
much  would  he  receive  for  25  carriages,  selling  them  at  the 
same  rate? 

5.  190  bushels  of  corn  cost  $100.70.     At  what  rate  must 
it  be  sold  to  gain  1 3  cents  a  bushel  ? 

6.  If  93  oranges  cost  $5.58,  what  will  75  oranges  cost? 

7.  If  12  yards  of  cloth  cost  $48.00,  what  will  7  yards 
cost? 

8.  If  16  horses  cost  $1952,  what  will  22  horses  cost  at' $6 
less  a  head  ? 

9.  A.  paid  $27,144  for  a  farm,  at  the  rate  of  15  acres  for 
$3510.     How  many  acres  did  he  buy? 

10.  If  46  acres  of  land  produce  2484  bushels  of  corn,  how 
many  bushels  will  120  acres  produce? 

INDICATED    SOLUTIONS. 

As  we  have  already  attempted  to  show,  the  solution  of  any 
problem  should  first  be  indicated  by  means  of  signs,  and 
afterwards  carried  to  completion  as  the  signs  direct. 

In  completing  a  solution  indicated,  a  parenthesis  or  a  vin- 
eulum  must  be  removed  first.  The  other  signs,  whether 
within  a  vinculum  or  not,  may  be  safely  used  in  the  following 
order:  X,  -=-,  — ,  -f. 

(12  H-  3)  X  2  =  4  X  2  =  8  ;  but  12  -=-  3  X  2  =  12  -*- 
6  =  2. 

5 


66  PRACTICAL  ARITHMETIC 

1.  Perform  the  operations  indicated  in  (48  X  2  —  84  -^  6  X 

2)  +  7  -  3. 

1.  (48  X  2  —  84  -r-  6  X  2)  -f-  7  —  3. 

2.  By  removing  sign  X,  (96  —  84  -+-  12)  -f  7  —  3. 

3.  By  removing  sign  -4-,  (96  —  7)  +  7  —  3. 

4.  By  removing  sign  — ,  89  -j-  4. 

5.  By  removing  sign  -f->  93. 

2.  Find  the  value  of  48  X  2  —  84  -*-  6  X  2  +  7  —  3. 

3.  If  6  men  can  do  a  piece  of  work  in  10  days,  how  long 
will  it  take  5  men  to  do  the  work? 

We  may  let  x  stand  for  the  required  number  of  days,  and  write  an 
equation  thus : 

Process.  Explanation. 

ANALYSIS. — Since  6  men  require  10  days, 
1  man  will  require  6  X   10  days.     Hence,  5 

x  -  r men  will  require  — — days.      Performing 

1 0  v  fi 

the  operations  indicated,  we  have  ^—  = 

x  =  12  days.  fi0  5 

J  2  =  12  days. 

5 

4.  If  12  men  can  build  a  school-house  in  25  days,  how 
long  will  it  take  25  men  to  build  it? 

5.  How  many  pounds  of  butter,  at  $.23  a  pound,  must  be 
given  for  5  pounds  of  raisins  at  $.11  a  pound,  2  pounds  of 
tea,  at  $.63  a  pound,  and  a  barrel  of  sugar,  at  $9  ? 

5  x  $.11  4-  2  X  $-63  4-  9 
Suggestion:  — ^- 

$.23 

(j.  Find  the  value  of: 

1.  28  X  6  ~  14  +  9  X  8  -r-  12  +  42  -f-  7  X  3. 

2.  99  X  (8  +  51)  X  10  —  (7  X  104  -f  26). 

3.  (105  -r-  21  +  80  -+-  5)  X  (81  +  36  -=-  9). 

16  X  3125  —  127  X  0  +  (380  -*-  (100  4-  50) 
239 


(125  X  30)  -4r  (25  X  25)  X  (32  —  21)  —  55  -f-  5 

n 


DIVISION  67 

7.  A  lady  paid  a  store  bill  of  $784,  giving  30  twenty- 
dollar  bills,  4  one-dollar  bills,  and  the  remainder  in  five-dollar 
bills.     How  many  five-dollar  bills  did  she  use  ? 

$784  —  (30  X  120  +  4  X  $1) 
$5 

8.  Two  trains  leave  New  York  for  Chicago,  900  miles,  at 
the  same  hour,  one  averaging  30  miles  an  hour,  the  other  45 
miles  an  hour.     How  long  will  the  second  train  be  in  Chicago 
before  the  first  arrives  ? 

9.  How  many  men  will  it  take  to  do  a  piece  of  work  in 
26  days  that  39  men  can  do  in  76  days  ? 

10.  How  long  can  125  men  subsist  on  an  amount  of  food 
that  will  last  3  men  4500  days  ? 

11.  A  quantity  of  provisions  lasts  an  army  of  2500  men  72 
days.     How  long  would  it  last  18,000  men  ? 

12.  I  bought  a  carriage  for  $140,  a  horse  for  $125,  and  a 
set  of  harness  for  $18;  kept  them  a  month  at  an  expense  of 
$17.25,  and  then  sold  the  team  for  $300.     Did  I  gain  or  lose, 
and  how  much  ? 

13.  A  pedler  sells  beets,  six  in  a  bunch,  at  10  cents  a  bunch, 
and  gains  one  cent  on  each  bunch.     Find  the  cost  per  C. 

Suggestion  :   10~  1  x  100  =  ? 
6 

14.  I  paid  $86.40  for  1440  blocks  of  granite.     What  was 
the  price  per  M  ? 

Suggestion:  ^~~  X  1000  =  ? 

15.  If  8  acres  of  land  cost  $656,  what  will  35  acres  cost  at 
$4  more  per  acre  ? 

16.  From  126  -f  (T6~+  4)  x  2  take  (48  -f-  2)  +  34  X  6 
-  (17  -  15). 


68  PRACTICAL    AKITHMETIC 

GENERAL   PRINCIPLES    OF   DIVISION. 

If  24  is  the  dividend  (D.),  4  the  divisor  (d.),  and  6  the 
quotient  (§.),  we  have 

4,  d. 

We  will  now  notice  the  effect  upon  Q ,  if  we  multiply  and 
divide  D.  and  d.  by  2?  as  follows  : 


2. 


4 


„ 

6. 


"  Analysis  reasons  from  particular  instances  to  general 
principles." 

Reasoning  from  the  particular  instances  above,  we  derive 
the  following 

PEINCIPLES. 

1.  Multiplying-  D.  multiplies  Q. 

2.  Multiplying  d.  divides  Q. 

3.  Dividing  D.  divides  Q. 

4.  Dividing  d.  multiplies  Q. 

5.  Multiplying  both  D.  and  d.  does  not  change  Q. 

6.  Dividing  both  D.  and  d.  does  not  change  Q. 

Let  D.  =±  1728  and  d.  =  144.  Find  §.,  and  illustrate  each 
of  the  above  six  principles. 


4 

24-=-  2 

4 

12    3  n 

Q  is  divided  by  2. 
Q.  is  divided  by  2. 
Q.  is  multiplied  by  2. 
Q.  is  unchanged. 
Q.  is  unchanged. 

4 

24 

24  -3 

4X2 
24 

8    - 
24        10 

2^22 

48        « 

24  -*-2 

12        fi 

4     •     '>. 

n 

DIVISION  69 

SHORT   PROCESSES   IN    DIVISION. 

When  there  are  ciphers  at  the  right  of  the  divisor,  the 
process  of  division  is  readily  simplified. 

The  Divisor  1  with  Ciphers  annexed. 

1.  Divide  539  by  10. 

Process.  Explanation. 

1  0  )  53J9  Cutting  off  the  digit  9  from  the  dividend,  and  the 

coin  0  from  the  divisor,  we  have  53  tens  -j-  1  ten  —  53, 

Quo.  Kern.  with  9  remaininS-     [Principle  6.] 

RULE. 

For  each  cipher  in  the  divisor  cut  off  a  digit  from  the 
right  of  the  dividend. 

2.  Divide: 

By  10.  By  100.  By  1000. 

1.  6327.  6.  3267.  11.  6173. 

2.  5327.  7.  5327.  12.  5432. 

3.  9732.  8.  9273.  13.  8650. 

4.  9267.  9.  5533.  14.  3000. 

5.  2567.  10.  1234.  15.  5678. 

The  Divisor  any  Significant  Figure  with  Ciphers  annexed. 

1.  Divide  7436  by  3000. 

Process.  Explanation. 

3|000  )  7  [436  Cutting  off  436  from  D.  and  000  from  o?.,  we  have 

2  1 4-^P  ^  tk°usancls  -*-  3  thousands  =  2,  with  one  thousund 

remaining.     1  thousand  -f-  436  =  1436.     Hence  Q. 
Quo.   Bern.  =  2;  and  R  =  1436 

Repeat  the  principle  involved. 

2.  Divide: 

1.  673  by  20.  5.  1074  by  80. 

2.  957  by  30.  6.  1096  by  90. 

3.  686  by  40.  7.  5736  by  200. 

4.  790  by  50.  8.  7300  by  300. 


70  PRACTICAL    ARITHMETIC 

3.  Complete  the  following  : 

1.  873  -*-  600  =  7.  10,432  •*•  4000  = 

2.  1052  -4-  700  -  8.  10,037  •*-  5000  = 

3.  1095  -4-  800  =  9.  9396  -*-  6000  == 

4.  1073  -4-  900=  10.  9116-r-  7000  = 

5.  5327  -h  2000  —  11.  10,370  -*-  8000  = 

6.  8645  -*•  3000—  12.  10,573  -5-  9000  - 

The  Divisor  any  Number  -with  Ciphers  annexed. 

1.  Divide  5658  by  3200. 

Process.  Explanation. 

32lOO^  56lo8  (  1  Quo-  Cutting  off  58  from  D.  and  00  from  d., 

we  have  56  hundreds,  quotient,  58  units  re- 
maining;  56  hundreds  -=-  32  hundreds  —  1, 
2458  Rem-  quotient,  with  24  hundreds  remaining;  24 

hundreds  -j-  58  units  =•  2458,  entire  remainder. 

2.  Find  the  value  of: 

1.  97,658  ~  3300  =  7.  500,896  —  11,000  = 

2.  59,625  -&.  4600  —  8.  485,432  —  23,400  = 

3.  78,695  •+•  5300  —  9.  306,959  —  30,500  = 

4.  89,765  -4-  4400  ==  10.  940,938  —  24,500  — 

5.  68,543  -4-  6400  =  11.  768,448  —  32,300  = 

6.  954,000  -4-  350  =  12.  533,337  —  38,000  = 

REVIEW. 

1.  Define  the  following  terms  : 

1.  Division.  7.  Long  Division. 

2.  Divisor.  8.  Short  Division. 

3.  Dividend.  9.  Analysis. 

4.  Quotient.  10.  Solution. 

5.  Remainder.  11.  Principle. 

6.  0  as  remainder.  1 2.  Parenthesis. 

2.  What  are  the  principles  of  division  ? 


DIVISION  71 

3.  In  a  solution  indicated  by  the  signs  you  have  learned  to 
use,  in  what  order  is  it  always  safe  to  use  these  signs  ? 

4.  Invent  five  problems  whose  solution  may  be  indicated 
by  five  different  signs. 


PROPERTIES   OF   NUMBERS. 

DEFINITIONS    AND   INDUCTIVE    STEPS. 

1.  A  Factor  (Latin,  "  maker")  of  a  number  is  one  of  the 
numbers  which,  multiplied  together,  produce  the  number,  as 
in  2  x  3  X  4  =  24. 

2.  Write  two  factors  that  will  produce  24.     Write  four 
factors  =  24. 

3.  Form  an  equation,  putting  five  factors  =  300. 

4.  An  Exact  Divisor  of  a  number  is  one  of  its  factors. 
What  are  the  exact  divisors  of  6? 

5.  Since  2  X  3  X  5  =  30,  are  2,  3,  and  5  factors  of  30, 
or  exact  divisors  of  30  ? 

6.  If  you   have  the  equation  2  X  ?  =  6,  how  can  you 
obtain  the  required  factor? 

7.  Since  2  X  3  X  ?  =  30,  how  can  you  obtain  the  re- 
quired factor? 

8.  Then  if  a  number  and  all  its  factors  are  given  except 
one,  how  do  we  find  that  one? 

9.  Has  2.  or  3,  or  5,  any  factors  except  itself  and  1  ? 

10.  A  number  that  has  no  factors  or  exact  divisors  except 
itself  and  one  is  a  Prime  number,  as  2,  3,  5,  7,  11,  etc. 

11.  A  number  that  has  factors  or  exact  divisors  other  than 
itself  and  one  is  a  Composite  number,  as  4,  6,  8,  9,  etc 

12.  The  Prime  numbers  between  1  and  100  are  as  follows  : 
2,  3,  5,  7,  11,  13,  17,  19,  23,  29,  31,  37,  41,  43,  47,  53,  59, 
61,  67,  71,  73,  79,  83,  89,  97. 


72  PRACTICAL  ARITHMETIC 

13.  All  the  other  numbers  between  1  and  100  are  called 
what?     [See  11.] 

14.  Why  are  they  so   named  ?      Ans. :    Because  they  are 
composed  of  factors. 

15.  12  =  3X4  is  a  correct  equation.    What  kind  of  factor 
is  3?     What  kind  is  4? 

What  are  the  two  equal  prime  factors  of  4?     Re- write  the 
equation  with  three  prime  factors  in  the  second  member. 

16.  An  Even  number  is  exactly  divisible  by  2. 

17.  An  Odd  number  is  not  exactly  divisible  by  2. 

18.  Is  3  an  exact  divisor  of  12  ?     Of  twice  12  ?     Of  three 
times  12?     Of  any  number  of  times  12  ? 

19.  3  is  an  exact  divisor  of  12  and  21.     Is  it  an  exact 
divisor  of  their  sum  ?     Illustrate. 

Is  it  an  exact  divisor  of  their  difference?     Illustrate. 

20.  Since  numbers  are  either  prime  or  composite,  factors  are 
either  prime  or  composite. 

PRINCIPLES. 

1.  Every  composite  number  is  the  product  of  its  prime 
factors. 

2.  Every  prime  or  composite  factor  of  a  number  exactly 
divides  that  number. 

3.  Every  exact  divisor  of  a  number  is  one  of  its  prime 
factors  or  the  product  of  two  or  more  of  its  prime  factors. 

4.  Every  exact  divisor  of  a  dividend  exactly  divides  any 
number  of  times  that  dividend. 

5.  A  common  divisor  of  two  numbers  or  dividends  ex- 
actly divides  their  sum. 

6.  A  common  divisor  of  two  numbers  or  dividends  ex- 
actly divides  their  difference. 

7.  Any  factor  of  a  number  becomes  a  quotient  when  the 
number  itself  becomes  a  dividend,  and  its  other  factor,  or 
the  product  of  its  other  factors,  becomes  a  divisor. 

SUGGESTION. — Pupils  should  be  required  to  illustrate  each  of  the  fore- 
going principles. 


PROPERTIES  OF  NUMBERS  73 

EXERCISES. 

1.  Write: 

1.  Three  prime  numbers  exceeding  100. 

2.  The  composite  numbers  between  75  and  100. 

3.  An   equation,  using  three  composite  numbers  as 

factors. 

4.  An  equation,  using  three  prime  numbers  as  factors. 
What  kind  of  number  is  the  second  member4?     Men- 
tion the  exact  divisors  of  it. 

2.  Write: 

1.  Three  even  numbers. 

2.  Three  odd  numbers. 

3.  The  prime  numbers  between  1  and  50. 

4.  The  prime  numbers  between  50  and  100. 

5.  The  single  even  prime  number. 

6.  Three  odd  numbers  that  are  not  prime. 

3.  Finish  the  following  equations  : 

1.  1  X  2  X  89  X  97  =  ? 

2.  3X5X7X31=? 

3.  47  X  ?  X  53  =  12,455. 

4.  ?  X  1  X  67  X  89  =  59,630. 

5.  41  X  43  X  47  X  0  =  ? 

4.  Why  are  not  49,  51,  and  63  prime  numbers? 


FACTORING. 

Factoring  is  the  process  of  obtaining  the  factors  or  exact 
divisors  of  a  number.  The  number  factored  is,  therefore,  a 
dividend. 

The  most  important  problem  in  this  connection  is  to  find 
the  prime  factors  of  a  number,  as  a  sure  means  of  obtaining 
certain  required  divisors  or  dividends. 


74 


PRACTICAL    ARITHMETIC 


EXERCISES 
FOR  ANALYTIC  AND  SYNTHETIC  EXPLANATION. 

1.   What  are  the  prime  factors  of  108  ? 


Process. 


2 
2 
3* 
3 

108 

54 

27 

9 

108  =  2  X  2  X  3  X 
3X3. 


Explanation. 

Since  the  prime  factors  of  a  number  are 
exact  divisors  of  the  number,  we  find  all  the 
prime  numbers  that  exactly  divide  108.  108 
being  an  even  number,  is  divisible  by  2 ;  54 
being  even,  is  divisible  by  2.  Dividing  by 
3,  and  again  by  3,  the  quotient  is  3,  a  prime 
number.  Hence  the  prime  factors  of  108  are 
2,  2,  3,  3,  3. 


NOTE. — In  finding  the  prime  factors  of  a 
number,  use  the  prime  numbers  as  divisors  in  order  of  their  values,  begin- 
ning with  the  lowest  one  that  will  divide  the  given  number. 


2.  What  are  the  prime 

factors  : 

1.  Of  72? 

17.  Of  168? 

33.  Of  798? 

2.  Of  35  ? 

18.  Of  231? 

34.  Of  484? 

3.  Of  64? 

19.  Of  178? 

35.  Of  1280? 

4.  Of  46? 

20.  Of  180? 

36.  Of  1898? 

5.  Of  336  ? 

21.  Of  144? 

37.  Of  5460? 

6.  Of  111? 

22.  Of  315? 

38.  Of  3420? 

7.  Of  385? 

23.  Of  420? 

39.  Of  1470? 

8.  Of  429  ? 

24.  Of  660? 

40.  Of  1492? 

9.  Of  925? 

25.  Of  740  ? 

41.  Of  2310? 

10.  Of  492? 

26.  Of  945  ? 

42.  Of  2772? 

11.  Of  1320? 

27.  Of  1728? 

43.  Of  1600? 

12.  Of  8424? 

28.  Of  4284  ? 

44.  Of  8364  ? 

13.  Of  7698? 

29.  Of  1682? 

45.  Of  2585? 

14.  Of  743? 

30.  Of  997  ? 

46.  Of  1997? 

15.  Of  3675? 

31.  Of  4620? 

47.  Of  4851  ? 

16.  Of  4536? 

32.  Of  5250? 

48.  Of  7623? 

PROPERTIES  OF  NUMBERS 


75 


3.  Find  the  composite  factors  of  40. 
Process. 


Prime 
factors. 


40^ 
20 


10 


Prime 

factors 
combined. 


1 


2X2  =4 

2X2x2=  8 
2X5  =10 
2  X  2  X  5  =  20 


Composite 
factors. 


1.  Of  102,    105,    108,    221,    715,    845. 

2.  Of  84,    250,    735,    9800,    11,165. 

3.  Of  231,    78,    415,    852,    452,    1227. 


REVIEW   OF    PRINCIPLES. 

[See  page  72.] 

1.  Factor  54,  and  illustrate  Principle  1. 

2.  Factor  36,  and  illustrate  Principle  2. 

3.  Factor  108,  and  illustrate  Principle  3. 

4.  Factor  144,  and  illustrate  Principle  4. 

5.  Factor  231  and  154,  and  illustrate  Principle  5. 

6.  Factor  360  and  320,  and  illustrate  Principle  6. 

7.  Factor  1728,  and  illustrate  Principle  7. 


MULTIPLICATION   BY   FACTORS. 

1.  What  will  24  carriages  cost  at  $257  each? 

Process. 

24  =  4  X  6 ;  $257  X  6  =  1542 ;  1542  X  4  =  $6168. 

2.  In  like  manner  find  the  cost  of : 

1.  35  cows  at  $53  each. 

2.  22  violins  at  $10.35  each. 

3.  72  cords  of  wood  at  $4.65  a  cord. 


76  PRACTICAL   ARITHMETIC 

4.  99  books  at  $2.18  apiece. 

5.  123  hats  at  $5.65  apiece. 

6.  51  acres  of  land  at  $125  an  acre. 

7.  49  barrels  at  $1.25  apiece. 

8.  63  bags  of  salt  at  $1.875  a  bag. 

9.  21  shot-guns  at  $55.50  apiece. 

10.  121  paper-weights  at  $.555  apiece. 

11.  132  horses  at  $132  a  head. 

12.  144  rifles  at  $87.50  apiece. 

13.  34  yards  of  silk  at  $2.56  a  yard. 

14.  81  bushels  of  wheat  at  $.95  per  bushel. 


DIVISION   BY   FACTORS. 

When  the  divisor  is  a  composite  number,  division  may 
sometimes  be  readily  performed  by  using  factors  of  the 
divisor. 

1.  Divide  3598  by  14. 

Process.  Explanation. 

3598  Tne  factors  of  14  are  2  and  7.    Dividing  3598  into  two 


1  7QQ  equal  parts,  and  each  of  those  2  equal  parts  into  7  equal 

— parts,  we  thus  obtain  7  times  2  or  14  equal  parts,  each 

257  equal  to  257. 

2.  Divide,  using  factors  : 

1.  8445  .*.  15.  7.  9345  -f-  105. 

2.  7776-^24.  8.  1152-4-72. 

3.  23,296  •+•  32.        9.  3648  -4-  96. 

4.  1152-4-64.  10.  42,336-^-49. 

5.  1855  -4-  35.  11.  37,464  -4-  42. 

6.  16,340^-38.  12.  153,160-^56. 

The  chief  difficulty  in  dividing  by  factors  is  to  find  the  true 
remainder.     Notice  the  following  explanation  : 


PROPERTIES  OF  NUMBERS  77 

3.  Divide  4753  by  140,  using  factors. 
Process.  Explanation. 
414753                              140  =  4  X  5  X  7. 

5    1188    •    •    1  unit  of  4753  remaining. 

237    .    .    8  units  of  1188  remaining  =  3  X  4  =  12  units  of  4753. 
33    .    .    6  units  of  237  remaining  =  6x5x4  =  1 20  units  of  4753 
1  -j-  12  -|-  120  =  133,  true  remainder. 

The  true  remainder  must  be  a  part  of  4753. 

Why  must  partial  remainder  3  be  multiplied  by  4  ? 

Why  must  partial  remainder  6  be  multiplied  by  5  X  4  ? 

4.  Divide,  using  factors  : 

1.  7304  by  24.  5.  2184  by  49. 

2.  4104  by  45.  6.  3824  by  32. 

3.  3276  by  27.  7.  3548  by  72. 

4.  3275  by  56.  8.  1299  by  56. 


CANCELLATION. 

Cancellation  abridges  the  process  of  division  by  striking 
out  a  common  factor  from  dividend  and  divisor. 

Striking  out  a  common  factor  is  in  effect  dividing  both 
dividend  and  divisor  by  the  same  number.  [State  the  prin- 
ciple, page  68.] 

1.  Divide  (96  X  9  X  8)  by  (12  X  16). 

Process.  Explanation. 

4  Cancelling  12  and  96,  we  have 

$  8  as  the  result  in  the  dividend ; 

<M  X  9  X  8  cancelling  8   and  16,  we  have  2 

12  V  7$ —  ==  ^  ^^  ~  ^'  as  the  result  in  *^e  divisor;  can- 

celling 2  and  8,  we  have  4  as 
the  result  in  the  dividend.  The 

entire  divisor  having  heen  cancelled,  the  quotient  is  the  product  of  the 

uncancelled  factors  4  and  9,  which  is  36. 


78  PRACTICAL   ARITHMETIC 

2.  Divide  4  x  2  X  8  X  21  by  36  X  8  X  2. 

Process.  Explanation. 

7  We  first  cancel  2  and  8  in  both  divi- 

fl  X  ff  X  )8  X  jffi  _     _7_  dend  and  divisor  ;  next,  4  and  36,  obtain- 

M  X  ft  X  $          ~  ~3  ing  9  in  the  divisor  ;  finally,  we  cancel  9 

a  and  21,  rejecting  from  each  the  factor  3, 

and  obtain  3  in  the  divisor  and  7  in  the 

dividend. 

3.  Divide,  using  cancellation  : 

1.  18  x  24  X  35  by  6  X  8  X  7. 

2.  30  X  10  X  9  X  4  by  8  X  5  X  6. 

3.  6  X  7  X  9  X  11  by  2  X  3  x  7  X  3  X  21. 

4.  10  X  6  X  84  X  42  by  12  X  5  X  24  X  7. 

5.  144  X  75  X  15  X  32  X  23  by  432  X  25  X  8  X  30. 

6.  400  X  18  X  30  x  42  by  270  X  20  x  30  X  14. 

7.  28  X  56  x  400  by  112  x  280. 

PROBLEMS. 

1.  A  farmer  exchanged  15  barrels  of  apples,  each  contain- 
ing 3  bushels,  at  $.80  a  bushel,  for  8  pieces  of  cloth,  each 
containing  30  yards.  Find  the  price  of  the  cloth  per  yard. 

Process. 
.10 
W  X  3  X  .ftft       3  x  .10       .30  _ 


fiXfiP  2  2 

2 

Explanation. 

The  farmer  had  15  times  3  bushels  of  apples  worth  15  times  3  times  $.80. 
He*  got  therefor  8  times  30  yards  of  cloth,  which  cost  him  as  much  per 
yard  as  8  X  30  is  contained  times  in  15  X  3  X  -80.  By  cancellation  we 
obtain  15  cents  as  the  cost  per  yard. 

2.  A  miller  bought  12  loads  of  wheat,  each  containing  130 
bushels,  at  $1.25,  and  gave  in  exchange  8  loads  of  flour  at 
$6.25  a  barrel.  How  many  barrels  were  there  in  a  load? 


PROPERTIES  OF  NUMBERS  79 

3.  I  exchanged  apples  at  $1 .50  per  bushel  for  25  days'  labor 
at  §1.20  per  day.     How  many  bushels  of  apples  did  it  take? 

4.  Three  pieces  of  cloth  containing  20  yards  each,  worth 
$5  a  yard,  were  exchanged  for  5  pieces  of  cloth  containing  40 
yards  each.     What  was  the  second  kind  of  cloth  per  yard  ? 

5.  How  many  pounds  of  coffee  at  24  cents  per  pound  are 
required  to  pay  for  3  hogsheads  of  sugar,  each  weighing  1464 
pounds,  and  worth  15  cents  per  pound  ? 

6.  Four  farms  containing  80  acres  each,  worth  $65  per  acre, 
were  exchanged  for  5  farms  containing  95  acres  each.     What 
was  the  value  per  acre  of  the  farms  received  in  exchange  ? 

7.  How  many  firkins  of  butter,  each  containing  50  pounds, 
at  18  cents  a  pound,  must  be  given  for  3  barrels  of  sugar, 
each  containing  200  pounds,  at  9  cents  a  pound  ? 

8.  If  25  Jersey  cows  each  give  8  quarts  of  milk  a  day,  at 
5  cents  a  quart,  how  many  pieces  of  matting  of  40  yards  each, 
at  50  cents  a  yard,  will  pay  for  the  milk  of  12  days? 

9.  A  tailor  bought  5  pieces  of  cloth,  each  piece  containing 
24  yards,  at  3  dollars  a  yard.     How  many  suits  of  clothes,  at 
18  dollars  a  suit,  must  be  made  from  the  cloth  to  pay  for  it? 

10.  A  grocer  bought  7  chests  of  Souchong  tea,  containing 
24  pounds  each,  at  $1.05  per  pound.     How  many  firkins  of 
butter,  at  $.35  a  pound,  will  be  required  to  pay  for  the  tea, 
each  firkin  containing  56  pounds? 

11.  I  bought  24  barrels  of  apples,  each  containing  2  bushels, 
at  the  rate  of  75  cents  a  bushel.     Find  the  number  of  cheeses, 
each  weighing  30  pounds,  at  15  cents  a  pound,  that  will  pay 
for  the  apples. 

12.  How  many  days'  work,  at  $1.80  a  day,  will  pay  for  84 
bushels  of  corn,  at  $.45  a  bushel  ? 

13.  If  52  men  can  dig  a  trench  in  15  days,  working  10 
hours  a  day,  in  how  many  days  will  25  men  dig  a  similar 
trench,  working  12  hours  a  day  ? 


80  PRACTICAL   ARITHMETIC 

COMMON  DIVISORS. 

INDUCTIVE    STEPS. 

1.  What  number  is  a  divisor  of  both  6  and  8  ? 
What  one  of  both  9  and  12? 

What  one  of  both  20  and  24  ? 

2.  Two,  three,  and  Jour  are  in  this  case  called  Common 
factors  or  Common  divisors. 

r    6  =  2  X  3  \ 

3.  Numbers  \     8=2x2X2^  prime  factors. 

1  20  =±  2  X  2  X  5  J 

What  single  prime  factor  is  common  to  all  the  numbers  ? 

What,  then,  is  the  common  divisor  of  6,  8,  and  20  ? 

What  prime  factor  is  common  to  8  and  20  only  ? 

Is  it  the  same  2  that  is  common  to  all  the  numbers,  or  is  it 
a  different  2  ? 

Is  there  a  2  that  is  not  common  to  any  two  of  the  numbers  ? 

What  two  other  factors  are  not  common  to  any  two  of  the 
numbers  ? 

f  15  =  3  X  5  -\ 

4.  Numbers  j  30  =  2  X  3  X  5  |  prime  factors. 

I  45  =  3  X  3  X  5  J  ' 

What  two  prime  factors  are  common  to  all  the  numbers  ? 
The  numbers  have  what  two  common  divisors  ? 
Is  the  product  of  3  and  5  a  common  divisor? 
State  the  principle. 
Is  15  the  greatest  common  divisor  of  15,  30  and  45  ?   Why  ? 

DEFINITIONS. 

1.  A   Common  Divisor  of  two   or  more   numbers   is  a 
number  that  exactly  divides  each  of  them. 

2 .  The  Greatest  Common  Divisor  (G.  C.  D.)  of  two  or  more 
numbers  is  the  greatest  number  that  exactly  divides  each  of  them. 


PROPERTIES  OF  NUMBERS  81 

3.  Numbers  that  have  no  Common  Divisor  are  said  to  be 
prime  to  each  other. 

PRINCIPLE. 

The  GK  O.  D.  of  two  or  more  numbers  is  the  product  of  all 
their  common  factors. 

EXERCISES. 

1.  What  is  the  G.  C.  D.  of  42,  56  and  70? 

Process.  Explanation. 

42  =  2  X  3  X  7  Resolving    the    given    numbers    into 

p.«  9  V   9   V   9  V  7  their  prime  factors,  we  have  42  =  2  X  3 

lU     *  Z  X  O  X   I  X7.     By  inspecting  these  prime  factors, 

2  X  7  =  14,  G.  C.  D.  we  find  that  2  and  7  are  the  only  prime 

factors   common  to  all  the  numbers.     Hence   their   product,  14,  is   the 
G.  C.  D.  of  42,  56  and  70. 

2.  Find  the  G.  C.  D.  of: 

1.  21,  35,  56.  5.  6,  12,  30. 

2.  12,  18,  24.  6.  15,  25,  30. 

3.  14,  35,  63.  7.  12,  18,  72. 

4.  9,  27,  36.  8.  105,  35,  70. 
An  abridgment  of  the  above  method  is  as  follows : 

Process.  Explanation. 

105    35    70  Dividing   by   the    common    prime 

^       ^ J^~  factors  5  and  7,  the  quotients  3,  1,  2 

are  seen  to  be  prime  to  one  another. 

"»     *">     *  Hence  5  and  7  are  all  the  factors  com- 

Q   J)   ==  5  y   7  ==  35  mon  to  all  the  numbers,  and  5  X  7  or 

35  is  the  G.  C.  D. 

What  is  meant  by  "prime  to  one  another"  f 

3.  Find  the  G.  C.  D.  of  the  following : 

1.  28,  42,  70.  5.  16,  48,  80. 

2.  84,  126,  210.  6.  84,  126,  210. 

3.  45,  105,  135.  7.  120,  240,  600. 

4.  60,  100,  200.  8.  44,  154,  110. 


82  PRACTICAL    ARITHMETIC 

9.  51,  105,  243.  17.   180,  300,  900. 

10.  36,  84,  132.  18.  360,  288,  720,  648. 

11.  36,  81,  135.  19.  290,  435,  232. 

12.  42,  54,  60.  20.  17,  27,  36. 

13.  75,  300,  450.  21.  30,  42,  63. 

14.  144,  576,  720.  22.  296,  407. 

15.  13,  91,  143.  23.  2121,  1313. 

16.  14,  98,  112.  24.  1326,  3044,  4520. 

Nos.  22,  23  and  '24  may  be  reserved  and  factored  by  the  next  process. 

"When  the  numbers  are  not  readily  factored,  a  method 
founded  on  principle  6,  page  72,  is  adopted. 

1.  What  is  the  G.  C.  D.  of  169  and  195? 

Process.  Explanation. 

169)195(1  By  Principle  6,  a  common  divisor  of 

-|  />q  169  and  195,  divides  the  remainder,  26, 

— — -  and  consequently  156  and  the  remainder 

13.     Since   13  exact'y  divides   itself  and 

156  26,  it  is  a  common  divisor  of  169  and  195. 

G.  C.  D.  —  13)  26  (  2  Tne  G-  C.  D    must  also  divide  26  and 

<yn  13,  and  since  it  must  divide  13  it  cannot 

exceed  13.     Therefore  13  is  the  G.  C.  D. 

of  169  and  195. 

2.  Find  the  G.  C.  D.  of : 

1.  187  and  209.  11.  1215  and  1878. 

2.  322  and  391.  12.  1071  and  1870. 

3.  186  and  217.  13.  3696  and  1440. 

4.  205  and  246.  14.  6237  and  3520. 

5.  329  and  423.  15.  333  and  592. 

6.  424  and  583.  16.  423  and  752. 

7.  488  and  671.  17.  697  and  820. 

8.  296  and  407.  18.  901  and  1060. 

9.  849  and  1132.  19.  3471  and  1869. 

10.  426  and  784.  20.  1584  and  2772. 


PROPERTIES  OF  NUMBERS  83 

When  it  is  required  to  find  the  G.  C.  D.  of  more  than  two 
numbers,  first  find  the  G.  C.  D.  of  two  of  them,  then  of  that 
G.  C.  D.,  and  one  of  the  remaining  numbers,  and  so  on  for 
all  the  numbers.  The  last  G.  C.  D.  will  be  the  G.  C.  D.  of 
all  the  numbers. 

3.  Find  the  G.  C.  D.  of : 

1.  492,  744,  1044.  6.   121,  181,  221,  241. 

2.  944,  1488,  2088.  7.  561,  6732,  1728. 

3.  216,  408,  740.  8.  630,  1134,  1386. 

4.  945,  1560,  22,680.  9.  462,  1764,  2562. 

5.  43,  473,  215,  344.  10.  7955,  8769,  6401. 

PROBLEMS. 

1 .  What  is  the  length  of  the  longest  chain  that  will  meas- 
ure exactly  the  length  and  the  width  of  a  field  484  rods  long 
and  420  rods  wide  ? 

2.  Three  fields  containing  24  acres,  18  acres  and  42  acres 
are  to  be  cut  each  into  the  least  number  of  smaller  fields  of 
equal  size.     Find  the  size  of  the  fields. 

3.  Two  vats  contain  respectively  7875  and  16,128  gallons. 
Find  the  cask  of  greatest  capacity  that  will  exactly  measure 
both  vats. 

4.  What  is  the  length  of  the  longest  pole  with  which  you 
can  measure  the  three  lengths,  132,  156,  and  168? 

5.  In  a  village  some  of  the  walks  are  56  inches  wide,  some 
70  inches,  and  others  84  inches.     What  is  the  width  of  the 
widest  flagging  that  will  suit  all  the  walks  ? 

6.  What  is  the  greatest  length  of  board  that  can  be  used 
without  cutting  in  fencing  a  triangular  field  whose  sides  are 
80,  112  and  144  feet? 

7.  The  Erie  Railroad  has  3  side-tracks  of  the  following; 
lengths :  3013,  2231,  and  2047  feet.     Find  the  length  of  the 
longest  rail  that  will  exactly  lay  each  side-track. 


84  PRACTICAL    ARITHMETIC 

8.  A  grain  dealer  has  2722  bushels  of  wheat,  1822  bushels 
of  corn  and  1226  bushels  of  beans  which  he  wishes  to  "  ship" 
in  the  smallest  number  of  bags  of  equal  size.     Find  the  size 
of  the  bags. 

9.  Find  the  size  of  the  largest  equal  packages  that  will 
contain  without  mixing  60   pounds  of  one  kind  of  tea,  75 
pounds  of  a  second  kind,  and  100  pounds  of  a  third  kind. 

10.  There  is  a  triangular  field  whose  sides  are  288,  450, 
and  390  feet.  What  is  the  least  number  of  rails  that  will 
enclose  it,  with  a  fence  5  rails  high  ? 


COMMON  DIVIDENDS. 

INDUCTIVE   STEPS. 

1.  In  the  expression,  30  -j-  5  =  6,  which  number  is  the 
dividend  ?     Which  the  divisor  ? 

2.  In  the  expression,  30  -h  6  =  5,  which  is  the  divisor? 

3.  Since  30  is  divisible  by  both  5  and  6,  it  is  called  a 
common  dividend  of  5  and  6. 

4.  Can  you  find  a  number  less  than  30  that  is  a  common 
dividend  of  5  and  6  ? 

5.  Since  such  a  number  cannot  be  found,  30  is  called  the 
Least  Common  Dividend  (L.  C.  Dd.)  of  5  and  6. 

6.  What  are  the  prime  factors  of  5  and  6?      2,  3,  5. 
What  are  the  factors  of  30  ?     2,  3,  5. 

7.  Hence  we  see  that  the  L.  C.  Dd.  of  two  or  more  num- 
bers is  composed  only  of  the  factors  of  those  numbers. 

8.  The  prime  factors  of  42  are  2,  3,  7.    Is  42  the  L.  C.  Dd. 
of  6  and  7?     Why?    Of  3  and  14?    Why?    Of  2  and  21  ? 
Why? 

9.  Does  42  contain  any  other  factors  than  those  of  6  and  7, 
3  and  14,  or  2  and  21  ? 


PROPERTIES  OF  NUMBERS  85 

(-6  =  2X3  1 

10.  Numbers  J8=2X2X2     [  prime  factors. 

I  20  =  2  X  2  X  5  J 

What  prime  factor  is  common  to  all  the  numbers? 
What  prime  factor  is  common  to  8  and  20  only  ? 
Is  this  2  a  different  2  from  the  other? 
What  three  prime  factors  are  not  common  to  any  two  of 
the  numbers  ? 

How  many  different  prime  factors  are  common? 

How  many  are  not  common  ? 

How  many  different  factors  in  all  ? 

Name  them. 

Which  one  is  common  to  all  the  numbers  ? 

Which  one  is  common  to  two  of  the  numbers? 

Which  three  are  not  common  ? 

11.  Three  classes  of  different  prime  factors  are  to  be  recog- 
nized :  (1)  Factors  that  are  common  to  all  the  numbers;  (2) 
Factors  that  are  common  to  some  of  the  numbers ;  (3)  Factors 
that  are  not  common  to  some  of  the  numbers. 

12.  Can  you  form  a  L.  C.  Dd.  of  two  or  more  numbers 
without  using  all  the  different  prime  factors  of  those  numbers  ? 
Why  not? 

A  factor  common  to  all  the  numbers  will  be  taken  how 
often  as  a  factor  of  the  L.  C.  Dd.  ? 

A  factor  common  to  some  of  the  numbers  will  be  taken 
how  often  as  a  factor  of  the  L.  C.  Dd.  ? 

Will  a  factor  not  common  to  some  of  the  numbers  be  used 
as  a  factor  of  the  L.  C.  Dd.  ? 

DEFINITIONS. 

1.  A  Dividend  of  a  number  exactly  contains  that  number. 

NOTE. — The  word  Multiple  has  commonly  been  used  instead  of  Divi- 
dend. 


86  PRACTICAL   ARITHMETIC 

2.  A  Common  dividend  of  two  or  more  numbers  exactly 
contains  each  of  them. 

3.  The  Least  common  dividend  (L.  C.  Dd.)  of  two  or 
more  numbers  is  the  least  number  that  exactly  contains  each 

of  them. 

PRINCIPLE. 

The  L.  C.  Dd.  of  two  or  more  numbers  equals  the  product 
of  all  the  different  prime  factors  of  the  numbers,  and  no 
other  factors. 

First  Method. 

What  is  the  L.  C.  Dd.  of  20,  30,  70  ? 

Process.  Explanation. 

Q^.  9  V  9  V   ^  ^e  ^rs^  res°lve  the  numbers  into  their  prime 

factors.     The  L.  C.  Dd.  equals  the  product  of 

=  2X3X5  all  thejr  different  prime  factors.      The  factors 

70  =  2  X  5  X  7  common  to  all  the  numbers  are  2  and  5.     The 

T     n  T\  i  factors  not  common  to  some  of  ihe  numbers 

L.  C.  -Dd.   -  ftre  2<  3.  and  7.     Hence  the  factors  of  the  L. 

5X7X3X2.  C.  Dd   are  2,  5,  2,  3,  and  7,  and  the  L.  C.  Dd. 

=  2x2x3x5x7  =  420. 

RULE. 

Resolve  the  given  numbers  into  their  prime  factors. 
Select  all  the  different  factors,  common  and  not  common, 
and  find  then*  product. 

EXERCISES. 
Find  the  L.  C.  Dd.  of: 

1.  28,  21,  14.  10.  30,  32,  36. 

2.  60,48,36.  11.  56,  72,  96. 

3.  28,  32,  64.  12,  20,  24,  36. 

4.  16,  24,  36.  13.  22,  33,  55. 

5.  15,  45,  60.  14.  36,  40,  48. 

6.  45,  30,  72.  15.  30,  50,  80. 

7.  20,  24,  27.  16.  25,  45,  75. 

8.  35,  40,  42.  17.  36,  48,  64. 

9.  80,  60,  200.  18.  100,  450,  900. 


PROPERTIES  OF  NUMBERS  87 

Since  a  common  factor  enters  but  once  into  the  L.  C.  Dd., 
an  abridged  method  is  usually  adopted. 

Second  Method. 
Process.  Explanation. 

6    20  2<  a  common  factor  of  8,  6,  and  20,  is  a 

factor  of  tho  L.  C.  Dd.      2,  a  common  factor 


4'  3'  -1  of  4  and  10,  is  a  factor  of  the  L.  C.  Dd.     2,  3, 


2?  3,     5  an(j  5^  jn  the  iast  jjne)  are  the  factors  that  are 

T     n    "HI      —Ov  not  common      Hence  L.  C.  Dd.  =  2  X  2  X 

LJ.  O.  1M.  -  2X3X5  =  120. 

Find  the  L.  C.  Dd.  of  the  following  : 

1.  15,  60,  75.  9.  12,  26,  52. 

2.  8,  12,  40.  10.  18,  24,  36. 

3.  14,  35,  56.  11.  12,  18,  24. 

4.  9,  18,  27,  54.  12.  18,  36,  72,  108. 

5.  27,  36,  45,  90.  13.  12,  48,  36,  70. 

6.  18,  21,  24,  27.  14.  17,  51,  34,  85. 

7.  $240,  §270,  §180,  §150.  15.  21,  24,  26,  28,  30. 

8.  9,  10,  14,  15,  18.  16.  45,  50,  60,  63,  84. 

Third  Method. 

The  two  foregoing  methods  of  finding  the  L.  C.  Dd.  show 
tLat  two  kinds  of  factors  come  into  play, — -factors  common 
and  factors  not  common. 

Since  the  G.  C.  D.  of  two  numbers  is  the  product  of  their 
common  factors,  the  quotients  of  the  numbers  divided  by  the 
G.  C.  D.  are  either  the  prime  factors  not  common  or  the  pro- 
ducts of  those  factors.  Hence,  to  find  the  L.  C.  Dd.  of  two 
numbers  not  readily  factored  : 

1.  Find  the  G.  C.  D.  of  the  numbers. 

2.  Divide  the  numbers  by  the  G-.  C.  D. 

3.  Find  the  product  of  the  Gr.  C.  D.  and  the  quotients. 


88  PRACTICAL  ARITHMETIC 

1.  Find  the  L.  C.  Dd.  of  849  and  1132. 

Process. 

(L)  (2.) 

849)1132(1  283)849,  1132 

849  3,     4 

G.  C.  D.- 283) 849 (3 
849 

(3.) 

L.  C.  Dd.  =  283  X  3  X  4  =  3396. 

2.  How  many  kinds  of  factors  does  the  L.  C.  Dd.  of  849 
and  1132  contain? 

3.  Why  did  we  first  find  the  G.  C.  D.  of  849  and  1132* 

4.  Why  did  we  divide  849  and  1132  by  283  f 

NOTE.— When  the  L.  C.  Dd.  of  more  than  two  numbers  is  required, 
first  find  the  L.  C.  Dd.  of  two  of  them,  and  then  the  L.  C.  Dd.  of  the 
result  and  a  third  number,  and  so  on. 

5.  Find  the  L.  C.  Dd.  of: 

1.  261  and  319.  15.  9797  and  10,403. 

2.  731  and  817.  16.  9523  and  11,663. 

3.  527  and  589.  17.  2479  and  3589. 

4.  91  and  117.  18.  3045  and  6195. 

5.  135  and  144.  19.  568  and  712. 

6.  169  and  221.  20.  11,023  and  6493. 

7.  357  and  612.  21.  1485  and  2160. 

8.  1417  and  1469.  22.  30,072  and  133,784. 

9.  17,640  and  18,375.  23.  9,144,407  and  10,347,059. 

10.  1110  and  777.  24.  1177,  1391,  1819. 

11.  4087  and  4757.  25.  2943,  2616,  4578. 

12.  1728  and  1898.  26.  31,  124,  217,  310. 

13.  321  and  314,259.  27.  113,  452,  1000,  1492. 

14.  7854  and  86,394.  28.  135,  144,  356,  612. 


PROPERTIES  OF  NUMBERS  89 

PROBLEMS. 

1.  What  is  the  least  number  of  oranges  that  can  be  di- 
vided equally  among  21,  24  or  30  boys? 

2.  Find  the  least  number  of  acres  in  a  farm  that  can  be 
divided  exactly  into  lots  of  12,  15  or  18  acres. 

3.  What  is  the  smallest  sum  of  money  I  can  consume  in 
paying  workmen  12,  14,  16,  18  or  20  dollars  a  week? 

4.  A  man  desires  to  purchase  a  piece  of  cloth  that  can  be 
cut  without  waste  into  parts  5,  6,  or  8   yards  long.     How 
many  yards  must  the  piece  contain  ? 

5.  How  many  bushels  does  the  smallest  bin  contain  that 
can  be  emptied  by  taking  out  7  bushels,  10  bushels,  or  30 
bushels  at  a  time  ? 

6.  If  5  boys  start  together  and  run  around  a  square  in  12, 
15,  16,  18  and  20  minutes  respectively,  in  how  many  minutes 
will  they  all  meet  at  the  starting-point  if  they  continue  their 
course  around  the  square  ? 

7.  How  many  quarts  does  the  smallest  vessel  hold  that 
can  be  filled  by  using  a  3-quart  measure,  a  4-quart  measure,  a 
5-quart  measure  or  a  6-quart  measure? 

8.  What  is  the  smallest  sum  of  money  that  can  be  wholly 
expended  in  buying  horses  at  $75,  cows  at  $50,  or  sheep 
at  $9? 

9.  What  is  the  shortest  piece  of  cord  that  can  be  cut  into 
pieces  10,  12,  15,  16  or  18  feet  long? 

10.  A  heap  of  pebbles  can  be  made  up  into  groups  of  25 ; 
but  when  made  up  into  groups  of  18,  27  or  32  there  is  in  each 
case  a  remainder  of  11.     Find  the  least  number  of  pebbles 
such  heap  can  contain. 

11.  What  is  the  smallest  number  that  can  be  divided  by 
360,  460,  636,  and  748,  respectively,  and  leave  a  remainder 
of  260  ? 


90  PRACTICAL   ARITHMETIC 

REVIEW. 

1.  Divide  5  X  15  X  80  X  56  X  91  by  10  X  5  X  16  X 
78. 

2.  Divide  18  X  15  X  90  by  12  X  27  X  25. 

3.  Find  the  greatest  number  that  will  divide  each  of  the 
two  numbers  849  and  1132,  and  explain  the  process. 

4.  What  is  the  L.  C.  Dd.  of  4,  9,  and  29  ? 

5.  How  many  barrels  of  sugar,  240  pounds  each,  at  5 
cents  a  pound,  can  be  exchanged  for  8  pieces  of  sheeting,  of 
45  yards  each,  at  1 0  cents  a  yard  ? 

6.  How  much  does  the  L.  C.  Dd.  of  1751  and  2369  ex- 
ceed their  G.  C.  D.? 

7.  Find  the  difference  between  the  G.  C.  D.  of  1,  3,  5,  7, 
9,  and  the  L.  C.  Dd.  of  2,  4,  6,  8,  10. 

8.  Find   the  least   number  of  oranges  that,  arranged  in 
groups  of  6,  7,  8,  or  9,  have  just  5  over  in  each  case. 

9.  I  have  just  money  enough  to  buy  a  whole  number  of 
dozens  of  oranges  at  $.40  a  dozen,  or  a  whole  number  of 
baskets  of  peaches  at  $1.25.     How  much  money  have  I? 

10.  Four  boys  start  together  to  run  around  a  square ;  the 
first  can  run  around  in  12  minutes,  the  second  in  15  minutes, 
the  third  in  16  minutes,  and  the  fourth  in  18  minutes.     How 
long  will  it  be  before  they  all  meet  at  the  starting  point? 

11.  Define  the  following  terms  : 

1.  Integer.  9.  Factoring. 

2.  Fraction.  10.  Cancellation. 

3.  Factor.  11.  Common  divisor. 

4.  Exact  divisor.  12.  Greatest  common  divisor. 

5.  Prime  number.  13.  Dividend. 

6.  Composite  number.    14.  Common  dividend. 

7.  Even  number.  15.  Least  common  dividend. 

8.  Odd  number.  16.  Prime  to  one  another. 


FRACTIONS  91 


12.  Repeat  the  principles  pertaining  to  Prime  and  Com- 
posite numbers ;  to  the  G.  C.  D. ;  to  the  L.  C.  Dd. 

13.  Repeat  the  rules  for  finding  the  L.  C.  Dd. 

14.  Invent  5  problems  that  will  involve  the  use  of  the 
L.  C.  Dd. 


FRACTIONS. 

INDUCTIVE    STEPS. 

1.  Divide  5  apples  into  2  equal  parts. 

5^-2  =  2,  with  one  apple  remaining  undivided.  To  indi- 
cate that  one  apple  remains  to  be  divided  into  2  parts,  we 
write  it  thus,  J,  and  call  the  expression  one-half.  The  exact 
quotient,  therefore,  of  5  -f-  2  is  2^,  read  "  two  and  one-half. 
|.  —  12.^  rea(J  "  One  and  two-thirds."  All  the  remainders, 
resulting  from  division,  are  commonly  written  over  their 
divisors,  and  thus  form  numerical  expressions  called  Frac- 
tions, because  they  denote  parts  of  a  unit. 

2.  It  is  the  divisor  that  names  the  fraction.     If  the  divisor 
is  2,  one  part  (-J-)  is  called  one -half ;  two  parts  (|)  two  halves. 
If  the  divisor  is  3,  one  part  (^)  is  called  one-third  ;  two  parts 
(^)  two-thirds. 

3.  In  like  manner  we  have  fourths,  fifths,  sixths,  secenths, 
eighths,  ninths,  tenths,  etc. 

f  is  read  "  five-fifths,"  and  is  equal  to  one.     Any  quantity 
divided  by  itself  gives  one  for  quotient. 

4.  .|  may  denote  one-half  of         One  whole. 

a   whole   line.     f  may  denote             one-half.                l 
two-thirds  of  a  line.  ' ^ 

In  this  way  we  might  illus-  1  2 

irate  fourths,  fifths,  sixths,  etc.  3  3 

2"'  i>  i>  G^c.,  are  calle(l  fractional  units. 


92  PRACTICAL    ARITHMETIC 

5.  f  denotes  how  many  fractional  units  ? 

6.  What  is  the  value  of  f  ? 

7.  Read  the  following  fractions :  },  f,  $,  -&,  Q,  fl  if. 
Which  has  a  decimal  divisor?     Which  is  equal  to  one? 

8.  Write  two-thirds,  four-ninths,  seven-twelfths,  ten-seven- 
teenths.    Write  a  fraction  whose  value  is  one. 

DEFINITIONS. 

1.  An  Integral  unit  is  a  whole  or  undivided  unit. 

2.  An  Integer  is  a  whole  unit  or  a  collection  of  whole 
units. 

3.  A  Fractional  unit  is  one  of  the  equal  parts  into  which 
the  unit  is  divided. 

4.  A  Fraction  is  a  fractional  unit,  or  a  collection  of  frac- 
tional units. 

As  we  have  seen,  a  fraction  is  the  expression  of  a  division 
that  cannot  always  be  performed,  and  is  written  with  the 
number  to  be  divided  (dividend)  above  a  horizontal  line,  and 
the  divisor  below  that  line. 

5.  The  number  below  the  line  is  called   Denominator, 
because,  as  we  have  seen,  it  names  the  fractional  unit. 

6.  The  number  above  the  line  is  called  Numerator,  be- 
cause it  numbers  the  fractional  units. 

7.  The  Numerator  and  Denominator  are  called  the  Terms 
of  the  Fraction. 

8.  A  Common  fraction   is  expressed   by  writing  both 
numerator  and  denominator,  as  in  -|,  J-J-,  ^f . 

9.  A  Decimal  fraction  is  usually  expressed  by  simply 
writing  a  point  before  the  numerator,  as  in  .5,  .37,  .25. 

10.  A  Proper  fraction  has  the  numerator  less  than  the 
denominator,  as  J,  -§-,  T9^,  etc. 

11.  An  Improper  fraction  has  the  numerator  equal  to  or 
greater  than  the  denominator,  as  -JJ-,  ^  ^,  etc. 


FRACTIONS  93 

12.  A  Mixed  Number  consists  of  an  integer  and  a  frac- 
tion, as  2^-,  7f,  etc.,  read,  "two  and  one-third,  seven  and 
three-fifths." 

EXERCISES. 

Analysis  examines  the  separate  parts  of  a  subject  and  their 
connection  with  one  another. 

1.  Analyze  J. 

9  is  the  denominator,  is  a  divisor,  makes  and  names  the  fractional  unit. 
7  is  the  numerator,  is  a  dividend,  and  numbers  the  fractional  units. 
|  is  a  proper  fraction,  its  terms  are  7  and  9,  and  its  value  is  less  than 
one. 

2.  Analyze  :  f  ,  f  f,  y,  f,  £,  y,  f,  f  ,  f,  ^  Hi  A,  ft, 


3.  Name  the  proper  and    improper   fractions  and  mixed 
numbers  among  these:  f,  3|,  fltf,  4  A,  f,  A,  A.  "t.  3W> 

A9o,  ¥,  H,  i»H,  W,  2?AV,  I*.  ¥>  i*.  3*- 

4.  Write  with  figures  : 

1.  Five-ninths.  11.  Sixty-five  hundredths. 

2.  Ten-elevenths.  12.  110  ninetieths. 

3.  Seven-twenty-firsts.  13.  211  eighths. 

4.  Six  and  two-thirds.  14.  Thirty  and  seven-eighths. 

5.  Seventy-eightieths.  15.  Five  twenty-fifths. 

6.  Ninety-one  ninetieths.  16.  Fifteen  -sixteenths. 

7.  314-tenths.  17.  Nine  thirtieths. 

8.  1898-millionths.  18.  Four  and  two-filths. 

9.  Ten  and  five-sixths.  19.  Three-thirds. 

10.  Ten-tenths.  20.  Two  and  twelve-  twentieths. 

5.  Write: 

1.  A  common  fraction.      3.  A  proper  fraction. 

2.  A  decimal  fraction.       4.  An  improper  fraction. 

5.  An  improper  fraction  equal  to  one. 
Point  out  the  terms  of  the  fractions  you  have  written. 


94  PRACTICAL    ARITHMETIC 

REDUCTION   OF   FRACTIONS. 

Reduction  changes  the  terms  of  a  fraction  without  changing 
its  value.  The  change  is  to  Higher  Terms,  to  Lower  Terms, 
or  to  Lowest  Terms. 

Reduction  of  Fractions  to  Higher  Terms. 

Is  not  $!  =  $£? 

May  we  not  multiply  the  terms  of  \  by  2  and  thus  obtain  f  ? 

PRINCIPLE. 

Multiplying  both  terms  of  a  fraction  by  the  same  number 
does  not  change  the  value  of  the  fraction.  (See  p.  68. ) 

NOTE. — The  pupil  should  perceive  that  fractions  are,  by  their  very 
nature,  subject  to  the  principles  of  division. 

EXERCISES. 

1.  Change  f  to  twentieths. 

Process.        Analysis.  Explanation. 

nrj    .     A f         ~L  =  2-Q-  ^ ne  division  shows  that  the  terms  of 

1  _     _g  the  fraction  must  be  multiplied  by  5  to 

5.  x  5  =  IA  change  fourths  to  twentieths.    Multiply- 

4"  =~  2T  ing  both  3  and  4  by  5  we  have  £J. 

RULE. 

Divide  the  required  denominator  by  the  given  denom- 
inator, and  multiply  both  terms  of  the  fraction  by  the 
quotient. 

2.  Reduce: 

1.  1$  to  60ths.  4.  Tf-0-  to  lOOOths. 

2.  ||  to  SOths.  5.  fj-  to  270ths. 

3.  £J-  to  40ths.  6.  ff  to  1  SOths. 


EEDUCTION   OF   FRACTIONS 

7.  ft  to  90ths.  14.  ±%  to  74ths. 

8.  ft  to  HOths.  15.  £1  to  42Dds. 

9.  f  to  99ths.  16.  if  to  38ths. 

10.  f  to  49ths.  17.  f  to  30ths. 

11.  J-  to  GOths.  18.  fo  to  lOOths. 

12.  f  to  24ths.  19.  f  to  lOtbs. 

13.  7   to  70ths.  20.  J&  to  12ths. 


Reduction  of  Fractions  to  Lower  Terms. 

Is  not  $£==$£? 

May  we  not  divide  both  terms  of  |  by  2  and  obtain  %  ? 

PRINCIPLE. 

Dividing  both  terms  of  a  fraction  by  the  same  number 
does  not  change  the  value  of  a  fraction.  (See  page  68.) 

EXERCISES. 

1.  Reduce  f|  to  eighths. 

Process.  Explanation. 

i  p  _._  o o  ^e  division  °f  1C  by  8  shows  that  both  terms  of  the 

\_       __  fraction  must  be  divided  by  2  to  change  sixteenths  to 

eighths.     Dividing  both  12  and  16  by  2,  we  have  f . 

RULE. 

Divide  the  given  denominator  by  the  required  denom- 
inator, and  divide  both  terms  of  the  fraction  by  the 
quotient. 

2.  Reduce: 

1.  |f  to  15ths.  6.  £fj  to  16ths. 

2.  if  to  9ths.  7.  UJ-  to  lOOths. 

3.  |4  to  lOths.  8.  ^t  to  12ths. 

4.  j|f  to  12ths.  9.  f|J  to  9ths. 

5.  to  4ths.  10.       i    to  949ths. 


96  PRACTICAL   ARITHMETIC 

Reduction  of  Fractions  to  Lowest  Terms. 

Eeduction  to  lowest  terms  requires  the  terms  of  the  fraction 
to  be  divided  by  their  greatest  common  factor  (G.  C.  D.). 

EXERCISES. 

Process.  1.  Reduce  fjf$  to  lowest  terms. 

1760)5280(3 

5280  Explanation. 

— 7T  The  G.  C.  D.  of  1760  and  5280  is  1760.     Di- 

viding  both   terras  of   the   fraction   by  1760   we 
176  orHflF  =  i  obtain  |,  the  lowest  terms.     (See  Principle,  p.  95.) 

RULE. 
Divide  both  terms  of  the  given  fraction  by  their  G.  C.  D. 

Continued  division  by  a  common  factor  will  secure  lower  or 
lowest  terms. 


4 )JL  ULQ.  —  4 )  440  —  11)110  10)10  .  J. 

4)5  280     4)1320  ~~  TT)"3~3~(F  ~~  1~0")30  —  3' 

The  terms  are  the  lowest  when  they  are  prime  to  each  other. 
2.  Reduce  to  lowest  terms  : 

00          (2-)  (3-)  (4.)  (5.)  (6.) 

o  rr  Q  Q  />  rj 


(70 

M 

(8.) 
it 

(9.) 

232 
3~7T 

(10.) 

m 

(11.) 

Ht 

(12.) 

(13.) 
42 

0  0 

(14.) 
If 

(15.) 

AV 

(16.) 

150 
180 

(17.) 

210 
2T2 

(18.) 

m 

(19.) 

Mi 

/  O  O 

(20.) 
II* 

(21.) 
Iff 

O  1  O 

(22.) 

m 

(23.) 

m 

(24.) 

m 

(25.) 

650 

780 

(26) 

1769 
192Q 

(27.) 

288 
'864 

(28.) 

648 
T2T 

(29.) 
T\82\ 

(30.) 

86  4^ 
1286 

REDUCTION    OF   FRACTIONS  97 

(31.)       (32.)          (33.)  (34.)  (35.)  (36.) 

726        1680          1694 
TtT2        T1T12  1848 


(37.)     (38.)     (39.)      (40.)      (41.). 

10605     6161      1710      5040      4692 
11445     7171    T4364     17160    T6 1 9  7  6 


Reduction  of  Integers  and  Mixed  Numbers. 

We  have  learned  that  -|,  f ,  f ,  or  f ,  etc.,  equal  one. 
How  many  halves  in  one  whole  thing  ? 
How  many  thirds  ?     Sixths  ?     Tenths  ? 
How  many  thirds  in  two  ?     In  2f  ? 

EXERCISES. 

1.  Reduce  8J  to  fourths. 

Process.  Explanation. 

8  =  -^-  Since  1  =  £,  8  =  -3?2- ;  and  8  +  f  =  ^  + 

8J  =  ^t  +  }  =  ^.         f  =  -V- 

BULB. 

Multiply  the  integer  by  the  denominator,  to  the  product 
add  the  numerator,  and  -write  the  sum  over  the  denom- 
inator. 

2.  Reduce: 

1.  61  to  fourths.  11.  15  to  fifths. 

2.  2£  to  thirds.  12.  13£  to  sixths. 

3.  121  to  halves.  13.  18^-  to  elevenths. 

4.  9f  to  sevenths.  14.  5^-  to  ninths. 

5.  16J  to  fourths.  15.  5^-  to  eighteenths. 

6.  13f  to  eighths.  16.  272^- to  elevenths. 

7.  31 4^-  to  twenty-firsts.  17.  278$  to  ninths. 

8.  673T82-  to  twelfths.  18.  946T%  to  thirteenths. 

9.  702|f-  to  elevenths.  19.  615f  to  fifths. 

10.  122TV  to  fifteenths.       20.  24 1  -fa  to  twenty-firsts. 
Have  your  results  been  proper  or  improper  fractions  ? 

7 


98  PRACTICAL  ARITHMETIC 

3.  Reduce  to  improper  fractions  the  following : 

1.  9|.         6.  223TV       11.  21  Of  16.  15Ty 

2.  17|.       7.  13f           12.  16&.  17.  108 

3.  28^.     8.  504f.         13.  62^-.  18.  51T37-. 

4.  27f.   9.  114TV   14.  159TV  19-  40 ff. 

5.  49f  10.  312fi   15.  67  20. 


Reduction  of  Improper  Fractions. 

How  many  dollars  in  $|-?     In  $-^8-  ? 

How  many  units  in  ±£-  ?    In  ^  ?    In  ^L  ?    Jn  ±/-  ?    In  &£•  ? 

What  kind  of  numbers  are  your  results? 

EXERCISES. 

1.  Reduce  ^^  to  an  integer  and  ^£  a  mixed  number. 
Process.  Explanation. 

£0-4-  =  72  Since  -^  indicates  the  division  of  504  by  7,  we 

50  5  __  YOI  divide  and  obtain  the  integer  72. 

RULE. 
Perform  the  division  indicated. 

2.  Reduce  the  following  improper  fractions  : 

1.  £|.        10.  ff          19.  4^6,          28. 

2.  ff        11.  |f.          20.  -^          29. 

3.  J$jL.      12.  ^I_L.        21.  *$-.          30.  3||ji 

4.  *&•      13-  H-          22«  W-          «-  Mtt- 

5.  ii.        14.  ||.          23.  -3^4-.          32. 

6.  ff.  15.  *&•  24-  W-  ;33- 

7.  -V5/.  16.  ^.  25.  *$?£-.  34. 

8.  ff  17.  -V/.  26.  ^f^.  35. 

9.  ||..  is.  ^|8,  27.  ^ffi.  36. 


REDUCTION    OF   FRACTIONS  99 

REDUCTION    OF    UNLIKE    FRACTIONS. 

1.  Have  \  and  f  like  or  unlike  fractional  units  ? 

2.  By  reduction  to  higher  terms,  \  equals  how  many  sixths  ? 
|  equals  how  many  sixths  ? 

3.  Are  £  and  £  like  fractions?     Why  ? 

4.  What,  then,  is  the  difference  between  Like  and  Unlike 
fractions  ? 

5.  Have  £  and  %  a  common  denominator? 

6.  What  is  the  least  common  dividend  of  the  denominators 
2  and  3  ? 

DEFINITIONS. 

1.  Like  fractions  have  the  same  fractional  unit. 

2.  Unlike  fractions  have  not  the  same  fractional  unit. 

3.  Like  fractions  have  a  Common  denominator. 

4.  Like  fractions   may  have  a  Least  common  denomi- 
nator.    (L.  C.  D.) 

PRINCIPLES. 

1.  A  common  denominator  of  two  or  more  fractions  is  a 
common  dividend  of  their  denominators. 

2.  The  least  common  denominator  of  two  or  more  frac- 
tions is  the  least  common  dividend  of  then:  denominators. 


EXERCISES. 
1.  Reduce  f  and  -J  to  like  fractions. 

Process.  Explanation. 

3  y  g  __  24  A  common  dividend  of  3  and  8  is  24 ;  there- 

fore 24  is  a  common  denominator  of  ^  and  |.     To 

4  r=  A  *  8  =  40  reduce  |  to  twenty-fourths  we  multiply  hoth  terms 
7          7x3  =  21  by  8 ;  to  reduce  |  to  twenty-fourths  we  multiply 
^  =~  ¥  x  3  =  2T           both  terms  by  3. 


100  PRACTICAL  ARITHMETIC 

2.  Reduce  |-,  -J,  and  ^  to  fractions  having  their  least 
common  denominator. 

Process.  Explanation. 

L.  C.  Dd.  of  6    9  r^^ie  least  common  denominator  of  the  frac- 

12  is  36  tions  is   tlie  least  common  dividend  of  their 

denominators.     The  L.  C.  Dd.  of  6,  9,  12  is  36. 

|-  —  |.  x  6  =  3JJ.  -yyre  therefore  multiply  the  terms  of  f  by  6,  the 

1_  __  7  x  4  =  2_8  terms  of  |  by  4,  the  terms  of  {\  by  3. 

HI  1   x   3  =  33  The  same  results  may  be  obtained  by  reason- 

ing thus:    Since  1  =  ff,  \=  &,  and  f  ==  fg. 
Reduce  the  two  other  fractions  in  like  manner. 

RULE. 

Find  the  L.  C.  Dd.  of  the  denominators,  divide  it  by  each 
denominator,  multiply  both  terms  of  each  fraction  by  the 
quotient  obtained  by  its  denominator. 

State  the  principle  involved.     (See  page  68.) 

Brief  directions  are : 

1.  Find  the  L.  C.  Dd. 

2.  Divide  by  the  denominators. 

3.  Multiply  the  numerators  by  the  quotients. 

4.  Place  the  products  over  the  L.  G.  Dd. 

Before  applying  the  rule  reduce  mixed  numbers  to  improper  fractions 
and  fractions  to  their  lowest  terms. 

3.  Reduce  -f-,  -|,  -^  to  like  fractions  having  their  L.  C.  D. 

Process. 
Introductory,  -f%  —  -J-. 

1.  L.  C.  Dd.  of  7,  8,  2  is  56. 

2.  ¥>¥>¥  =  8,  7,  28. 

3.  3  X  8,  7  X  5,  1  X  28  =  24,  35,  28. 

4.  |f,  H,  H . 

4.  Reduce  in  similar  manner  the  following  : 

i.  i  i  I-  3-  T>  A.  W- 

2.  i  i  A-  4.  |,  f ,  f . 


ADDITION   OF   FRACTIONS  1 01 

5.  f,f,ff.  14.  3f,i  7,  11 

6.  f,  |,  ||.  15.  91  f ,  A,  |. 

7.  f,  T9o,  if-  16.  2J-,  4$,  4,  f 

8.  },  f,  A-  17.  8,  ?i,  f,  f 
9-  T82>  H,  TV  18.  i,  t,  i,  t,  *• 

10.  |,  f,  2f  19.  A,  H,  A;  iWr, 

11.  81  21  41  20.  A,  6J,  T9I7>  7,  f, 

12.  f,  A,  H.  21.  if,  if,  f 

13.  tt,A,«-  22.  i,  A,  A, 


ADDITION  OF  FRACTIONS. 

INDUCTIVE   STEPS. 

1.  What  is  the  sum  of  2  books  and  3  books? 

2.  What  is  the  sum  of  f  and  f  ? 

3.  Of  f  and  |?     Of  -^  and  ^? 

4.  What  is  the  fractional   unit  of  ^-?    Of  ^T?     Of  •&? 
Are  these,  then,  like  or  unlike  fractions  ? 

5.  What  kind  of  fractions  can  be  added  ? 
6._Can  you  directly  add  -f^  and  T6^-? 

7V  If  these  fractions  kad  a  like  or  common  denominator, 
could  you  add  them  ? 

8.  How  do  you  reduce  unlike  fractions  to  like  fractions? 

PRINCIPLES. 

1.  Only  like  fractions  can  be  added. 

2.  Unlike  fractions  can  be  reduced  to  like  fractions  and 
then  added. 

EXERCISES. 

,  1.  Find  the  sum  of  ^-,  y7^  and  -f^. 

Process. 


State  the  principle  involved. 


102  PRACTICAL   ARITHMETIC 

2.  Find  the  sum  of  f  ,  f,  f  . 

Process. 
1.  L.  C.  Del.  of  8,  6,  9  =  72. 

2.  ¥,¥,¥  =  9>  12>  8- 

3.  3  X  9,  5  X  12,  4  X  8  =  27,  60,  32 

4.  »T  +      " 


Explanation. 

Since  the  fractions  are  unlike,  we  render  them  like  by  reducing  them 
to  fractions  having  the  L   C.  D.  72. 

t  +  I  +  t  =  H  -f  ff  +  ft  =  W  =  iff 


3.  Find  the  sum  of  4J,  3J,  4|  and 
Process.  Explanation. 

Introductory.  -A-  =  -I- 

"  The  numbers  to  be  added  are  composed 

"?  ~       :    '    "2"IT  of  integers  and  fractions.     We  therefore 

3^-  =  3  -j-  -^j-  add  the  integers  and  fractions  separately, 

41  —  4  _j_  JJL  and  then  unite  their  sums.     4  -f  3  -f  4  -f 

5  =  1  6.     After   reduction   to   twentieths 


the  sum  of  the   fractions  is  |§,   or 
16  -f  1&  =  17230-,  the  sum  total. 


RULE. 

1.  Reduce  the  fractions,  giving  them  a  common  denomi- 
nator. 

2.  Add  the  integers  and  the  fractions  separately,  and 
unite  their  sums. 

4.  Find  the  sum  of  the  following  : 

1.  *,*,A-  7.  3|,  4J,  If,  2. 

2.  4fc  3J,  4*,  5fL  8.  H.  A.  10,  ff. 

3.  i,  f  ,  f  ,  I-  9.  A,  A,  »f 

4-  I,  f  ,  i,  t,  H-  10-  H-  H.  «• 

5-  A,  A,  A-  H-  »•  *i.  2i.  3f,  7i,  HI- 
6.  8^,  lOf,  14|,  12.  |,|,A»i 


ADDITION   OF   FRACTIONS  103 

o.A.A-         17.6i4,|,8. 

14.  A,  fV,  ft,  if  18.  15fcl7f,f 

15.  8  ft,  6T8f,  514,  H-     19-  900rV,  450f,  75^. 

16.  f,  H.  A,  A,  it-      20-  f  i  i,  t.  i  f  i- 

5.  What  is  the  value  of: 

1-  l  +  t  +  r?  6.  5f  +  18^  +  25^? 

2-  f  +  I  +  if?  7-  187i  +  ly7*  +  746?? 
3.  4i  +'  3f  +  H?  8-  17(54I  +  7867I  +  /T- 
4-  f  +  f  +  i  +  fl?  9.  211  +  331  +  6-Z&  +  7 

5-  A  +  ^  +  l+Vr?   10.  f  +  i 

6.  Answer  the  following  inquiries  ; 

1-  A  +/f  +"!+*=? 

2-  A-HT73-HA^? 

».-A  +  A-f*=? 

4.  187^  +  1976|-f  7461=? 

5.  8|+9|-f  12^=? 

6-  7H  +  8A  +  9*f  =? 

7.  ll|  +  10rV 

8.  81  +  61  +  2^. 

9.  51  +  6f  +  711 

10.  9}  +  lOf  +  llf  +  51J  .+  7T8r  +  18f  =? 


PROBLEMS. 

1.  I  bought  3  pieces  of  cloth  containing  125^,  96f,  and  48-| 
yards.     How  many  yards  in  the  three  pieces  ? 

2.  A  merchant  sold  a  customer  22^-  yards  silk,  3J  yard, 
paper  muslin,   11  yards  silesia,   5f  yards  cambric,  and   5^ 
yards  ruffling.     How  many  yards  were  sold  ? 

3.  A  farmer  divides  his  farm  into  5  fields.     The  first  con- 
tains 26-^  acres,  the  second  40^-f-  acres,  the  third  5  If  acres, 
the  fourth  59^  acres,  and  the  fifth  62|-  acres.     How  many 
acres  in  the  farm  ? 


104  PRACTICAL   ARITHMETIC 

4.  A  bicycler  rode  27f  miles  on  Monday,  33^  miles  on 
Tuesday,  37f  miles  on  Wednesday,  and  42|  miles  on  Thurs- 
day.    How  far  did  he  ride  in  the  four  days  ? 

5.  A  dry-goods  merchant  sold  a  lady  18 J  yards  of  flannel, 
21-J  yards  of  silk,  and  as  many  yards  of  calico  as  of  both 
the  other  goods.     How  many  yards  in  all  did  he  sell  ? 


SUBTRACTION   OP   FRACTIONS. 

INDUCTIVE   STEPS. 
1.   From  |-  subtract  -J. 

2-  A  —  A  =  what? 

3.  If  you  have  $-J  (of  a  dollar)  and  spend  $-|,  how  much 
have  you  left? 

4.  If  you  have  $-J  and  spend  $J,  how  do  you  find  the 
remainder  ? 

5.  What  kind  of  fractions  can  be  subtracted  without  re- 
duction. 

6.  What  kind  require  reduction  ?     Reduction  to  what  ? 

7.  Give  four  brief  directions  for  such  reduction. 

8.  What  introductory  step  is  sometimes  necessary  ? 

PRINCIPLES. 

1.  Only  like  fractions  can  be  subtracted. 

2.  Unlike  fractions  can  be  reduced  to  like  fractions  and 
then  subtracted. 

EXERCISES. 

1.  Find  the  difference  between  -^  and  ^-. 

Process.  Explanation. 

Since  T8T  and  T5T  are  like  fractions,  having  a 

Ijf  —  -ff  —  IT  common   denominator,    11,    their  difference   is   8 

elevenths  —  5  elevenths,  or  3  elevenths. 


SUBTRACTION   OF  FRACTIONS  105 

2.  What  is  the  difference  between  ^-  and  f  ? 

Process.  Explanation. 

Since  T\  and  f  are  unlike  frac- 

2          45          32--13  tions,  we  reduce  them  to  eightieths, 

tV  ~  ""80"         "SIT  making  them  like  fractions.     -^  —  f 

=  If-  ft  =  tt- 

3.  Subtract  7|  from  llf. 

Process.  Explanation. 

We   subtract  integers   and   fractions   sepa- 
^  8  rately.     f  cannot   be   taken   from   f ;    but    1, 

7|  =  _7j[_  taken  from  11,  equals  f  ;  f  +  f  ==  -^  5  ¥  -  I 

o£   03  =  |  or  |.     10  —  7  =  3.     Uniting  the  two  re- 

sults, we  have  3|,  the  remainder. 

RULE. 

1.  Reduce  unlike  fractions  to  a  common  denominator. 

2.  "Write  the  difference  of  the  numerators  over  the  com- 
mon denominator. 

3.  Subtract  integers  and  fractions  separately,  and  unite 
the  results. 

4.  From  f  take  |.     From  f  take  £. 

5.  From  f  take  £.     From  f  take  f . 

6.  From  |  take  ^-.     From  ft  take  ^. 

7.  From  |f  take  ^.     From  ££  take  ^-. 

8.  From  f  take  -3^.     From  fj  take  ^. 

9.  From  lOJf  take  if.     From  112  take  75f 

10.  From  606f  take  70J.     From  506|  take  418f 

11.  What  is  the  value  of: 

l-ff-ff-  7.  1198|-149|. 

2-  m  -  A-  8-  589|  -  67|. 

9. 

10.  72 

II. 
6-  ^V  -  T£T-  12.  42 


106  PRACTICAL   ARITHMETIC 

12.  Find  the  value  of: 

1.  51— 20ft.       8.  48ft  — 22£  15.76^- 

2.  66  —  36^.       9.  35f  —  29f  16. 

3.  64  —  59 f.       10.  44|  — 27|.  17. 

4.  38  —  37^.     11.  48f  —  9.  18.   loif  — 

5.  59  —  32-J-f     1 2.  73-i-  —  27|.  1 9.  28|f  —  1 6|f 

6.  Ill— 31|f   13.  22|  — 7f  20.  56^—  29f 

7.  36f  —  27f     14.  88T4T  —  53^.  21.  65|J  —  30f 

13.  Answer  the  following  inquiries  : 

3.  2i|-2^  ' 

4K      Q__     _ 
•          10 

5.  13^4 

6.  9|- 

8.  -^Q-  — 

9.  3f  +  9^  +  6|  =  ? 

10.  36  —  21  —  4|  —  61  —  ? 

11.  20  — 84  — 6ft  — f=? 

12.  200  —  30f  —  17^  —  26|f  ==  ? 


14.  53-^-21-9^  =  ? 


The  teacher  will  suggest  the  shortest  method  of  answering  the 
above  inquiries. 

PROBLEMS. 

1  .  3  \  yards,  4f  yards,  and  1  2^  yards  were  cut  off  from  a 
piece  of  silk  containing  30£  yards.  How  many  yards  re- 
mained ? 

2.  A  man  spent  ^  of  his  income  for  rent,  ^  for  food,  and  ^ 
for  other  expenses.  What  part  of  his  income  remained  ? 


MULTIPLICATION   OF   FRACTIONS  107 

3.  A  farmer  sold  -J  of  his  corn  to  one  man,  f  to  another, 
and  had  50  bushels  remaining.     How  much  corn  had  he  at 
first? 

4.  Show  that  the  fraction  fj-  is  greater  than  £  and  less  than  f  . 

5.  If  I  pay  my  grocer  $18  j,  my  coal  dealer  $271,  and  my 
tailor  $22|,  how  much  will  I  have  left  out  of  four  20-  dollar 
bills? 

6.  ^  of  a  pole  is  in  the  mud,  ^-  of  it  is  in  the  water,  and 
the  rest  of  it  is  in  the  air.     What  part  of  it  is  in  the  air  ? 

7.  Show  that  13f  —  2^  —  6-ft  +  3  —  1T%  +  8|  —  f  f 
—  ]  o^|  =  4f-|  is  a  correct  equation. 

8.  Find  the  second  members  of  these  ; 

1.  450  +  (12  X  5)  —  86fo  — 

2.  gi 

3.  59  —   2 


4.  52£  -f  (87  — 

5.  231  —  62|  +  101^-  —  |  =  ? 

6.  453  —  (32-^  +  f  —  10)  =  ? 

7.  LXXVII.  —  iV  +  CLXIX.  —  11^.  =? 


MULTIPLICATION  OF  FRACTIONS. 

INDUCTIVE  STEPS. 

1.  How  much  is  2  times  3  dollars? 

2.  How  much  is  2  times  3  sevenths? 

3.  How  much  is  2  times  ^-? 

4.  How  much  is  5  times  |-? 

5.  Multiply  ^  by  10.     •&  by  3. 

6.  Have   you   been   multiplying   numerators   or  denomi- 
nators ? 

7.  Then  what  effect  has  multiplying  the  numerator? 

8.  What  is  3  times  f  ?     What  are  the  lowest  terms  of  f  ? 


108  PRACTICAL  ARITHMETIC 

9.  Then  3  times  -|  =  |.     How  could  you  have  obtained 
|  more  directly  than  by  multiplying  the  numerator  ? 

10.  What  effect,  then,  has  dividing  the  denominator? 

11.  Is  that  effect  in  agreement  with  principle  4,  page  68  ? 
Why? 

PRINCIPLE. 

Multiplying  the  numerator  or  dividing  the  denominator 
multiplies  the  fraction. 

EXERCISES. 

1.  Multiply  Jg-  by  4. 

Process.  Explanation. 

7     vx  4  __  _T_  According  to  the  principle  we  may  multiply  the 

numerator  or  divide    the   denominator.      Since    the 
denominator,  16,  is  divisible  by  4,  we  divide  and  obtain  the  result,  |. 

2.  Multiply  4  by  -fr. 

Process.  Explanation. 

4  X     77  =  —  —  111  Since  the  denominator,  17,  is  not  exactly 

divisible  by  4,  we  multiply  the  numerator 
by  4  and  obtain  the  result,  \ f  —  1||. 

RULE. 

To  find  the  product  of  an  integer  and  a  fraction  divide 
the  denominator  or  multiply  the  numerator  by  the  integer. 

3.  Multiply: 

1.  A  by  7.  8.  A  by  4.  15.  if  by  18. 

2.  A  by  5.  9.  #  by  11.  16.  #  by  6. 

3.  if  by  7.  10.  fj  by  14.  17.  j||  by  18. 

4.  JL.  by  6.  11.  T3T  by  5.  18.  ^  by  10. 

5.  ^  by  8.  12.  -fa  by  3.  19.  -fa  by  28. 

6.  U  by  3.  13.  if  by  9.  20.  -^  by  19. 

7.  l|  by  13.  14.  #  by  14.  21.  -^  by  12. 


MULTIPLICATION   OF  FRACTIONS  109 

4.  Multiply  8f  by  4. 

Process. 

Explanation. 

4  times  f  =  f  =  2£. 
4  times  8  =  32. 
2-J-  32  +  2£  =  34£. 

34^  5.  Find  the  value  of: 

1.  9|  X  6.         8.  28|f  X  10.  15.  45ff  X  60. 

2.  7f  X  9.         9.  |fj  X  48.  16.  19£f  X  14. 

3.  8^  X  5.     10.  -fffs  X  144.  17.  25^|  X  15. 

4.  18$  X  8.     11.  18|  X  10.  18.  46|f  X  13. 

5.  21f  X  4.     12.  9f  X  21.  19.  54f|  X  35. 

6.  6|  X  13.     13.  8f  X  24.  20.  65|f  X  68. 

7.  171  X  9.     14.  63f  X  56.  21.  77TVr  X  77. 
6.  Multiply: 

1.  9  by  ^  8.  100  by  ^.  15.  $406  by  T3T. 

2.  57  by  |f.         9.  144  by  |f  16-  *718  b7  if 

3.  88  by  |.         10.  51  by  -^.  17.  $825  by  f|. 

4.  17  by  58T.       11.  75  by  $f.  18.  $fff  by  49. 

5.  12  by  if.       12.  90  by  ^.  19.  $^|  by  26. 

6.  124  by  -jV     13.  91  by  ^J-.  20.  $400  by  f. 

7.  153  by  ^.     14.  $318  by  ^-.  21.  $|-fi  by  f 


STEPS   TO   GENERAL   RULE. 

1.  Both  expressions,  3  X  4  and  4  X  3,  =  ? 

2.  What  principle  do  you  find  established  on  page  37  ? 

3.  How  much  is  -J-  X  6  ?     ^XlS? 

4.  How  much  is  6  X  |?     18  X  i? 

5.  How  much  is  ^  of  6  ?     \  of  18  ? 

6.  "Of"  between  a  fraction  and  a  following  number  is 
equivalent  to  what  sign? 

7.  Express  27  X  |  by  using  "of." 


110  PRACTICAL   ARITHMETIC 

8.  In  how  many  ways  can  you  indicate  the  product  of  16 
and  |? 

PRINCIPLES. 

1.  Fractions,  as  factors,  may  be  used  in  any  convenient 
order. 

2.  A  fractional  multiplier  may  be  used  as  expressing  the 
part  of  the  multiplicand  to  be  taken. 


EXERCISES. 

1.  Multiply  f  by  f. 

Process.  Explanation. 

lof|  =  ^  |X  |  =  |  of  |,  Principle  2.     ioff  =  F£7      f  of  f 

4  —  1£!  =  ^.     See  Principle,  page  68. 

4 

GENERAL  RULE. 

"Write  the  integers  and  mixed  numbers  in  fractional 
form;  cancel  common  factors,  and  find  the  product  of  the 
remaining  factors  of  the  numerators  for  a  new  numerator, 
and  of  the  denominators  for  a  new  denominator. 

2.  Find  the  value  of: 

1.  fofiJ.  6.*  of  AX  A  of  ft- 

2.  «•  of  ».  7.  |f  of  |f  X  U  of  «- 
3-  tfoftf  8.#ofWof&of#. 

4.  «of|f  9.  foffoff  Xfoff£. 

5.  ttofffr.  10.  fXAofHoff. 

3.  Reduce: 

1.  f  X  31  5.  Jfr  X  f  9.  »  X  3|. 

2.  AX  12f  6.  «X  A-  10.  lAXltt- 

3.  91  X  f  7.  A  X  f.  11.  «  X  A. 

4.  2^  X  If  8.  %  X  2f.  12.  f  x  #  X  6f 


MULTIPLICATION    OF   FRACTIONS  11] 

4.  Reduce  : 

1.  f  of  If  X  £  of  ^  of  4. 

1  &X  4r  X  &.x  f  x  tf 

3-  ttx^xAx^xtt- 
4.  Hx^xttxiixf 
&  A  x  H  x  tf  x  f  x  &. 
«•  »  x  &  x  tf  x  A  x  If 

'  •   1T6"  X  T5"  ^  T2TF  X   9-3   X  TTT* 

8.  f  x  |  x>  x  $  x  f  x  f 

9-AxAx-fxlxfxM- 
10.  *  x  «  x  If  x  #  x  &  x  f 

5.  What  is  the  value  of  : 

1.  f  of  f  of  5  X  fVof  |  of  3£? 

2.  |  of  ^  of  8  X  f  of  ^  of  15? 

3.  3^  X  f  X  4  X  f  of  7  ? 

4.  5^  times  ^  X  18  X  f  of  3  times  |  of  4? 

5.  &  of  15  X  |  of  -^  of  ^  of  6? 

6.  ^of  3f  of  if  X  |of  49? 

7.  if  of  if  XH  of 
8-        o 


PBOBLEMS. 
Required  the  cost  of: 

1.  45  pairs  of  shoes  at  $1  J-  per  pair. 

2.  -^j-  of  a  yard  of  cloth  at  $|-  a  yard. 

3.  1  20  yards  of  ribbon  at  1  6f  cents  a  yard. 

4.  4f  tons  of  hay  at  $16|  per  ton. 

5.  465  Rochester  lamps  at  $7-|-  apiece. 

6.  250  tons  of  coal  at  $6|  a  ton. 

7.  12^  cords  of  wood  at  $5f  a  cord. 

8.  If  16^  feet  make  a  rod?  how  many  feet  are  there  in 
rods? 


112  PRACTICAL    ARITHMETIC 

9.  There  are  24f  cubic  feet  in  a  perch  of  stone.     How 
many  cubic  feet  in  5^-  perches  ? 

10.  Mr.  Lipmann  bought  a  lot  of  crockery,  of  which  the 
retail  price  was  $576-|,  but  he  got  a  reduction  of  ^  for  whole- 
sale and  ^  for  cash.  What  amount  did  he  pay  ? 


DIVISION  OP  FRACTIONS. 

INDUCTIVE    STEPS. 

1.  1  divided  by  1  equals  what? 
4  divided  by  1  equals  what  ? 
-£  divided  by  1  equals  what  ? 
-1-  divided  by  1  equals  what  ? 

2.  If  -|  ~  I  =  -J,  |-  -f-  ^  equals  how  many  times  -J-  ? 

3.  If  -J-  -7-  -J-  =  3  times  -J-,  •§•  -r-  -§-  =  f  times  -J-.     Hence, 
-l==iX|or|of  1 

4.  Divide  in  like  manner  -J  by  |^  and  -|-  by  ^., 
What  principles  on  page  68  cfo'c/  yo?/,  apply  f 

What  change  in  the  form  of  the  divisor  do  you  observe? 


EXERCISES. 
1.  Divide  f  by  f. 

Process.  Explanation. 


become  X,  and  |  has  become  |,  i.e.,  has  become  inverted. 

2.  Divide  4  by  £. 

Process.  Explanation. 

4_i_7.  —  4.><8=J[2  —  44  4  =  f-     Inverting  |  and  writing 

sign  X,  we  have  f  X  f  =  -3/  =  4f 


DIVISION   OF  FRACTIONS  113 

3.  Divide  T97  by  3. 

Process.  Explanation. 

3  3 

_^-_?-l_A  -  +  3  =  lof9-  =       x      which  by 

~ 


10    '  10        3        10 

cancellation  gives  T3^. 

RULE. 

1.  Give  integers  and  mixed  numbers  fractional  form. 

2.  Invert  all  divisors. 

3.  Cancel  factors  common  to  numerators  and  denomi- 
nators. 

4.  Find  the  product  of  the  remaining  factors. 

4.  Divide  : 

1.  15  by  f  2.  18  by  f  6.  75  by  |£. 

•  Process.  3.  63  by  f  7.  32  by  f  . 

'  4.  25  by  f  8.  45by«. 

f  X  |=  21        5.  49  by  £  9<  64  bv  i^ 

5.  Divide  : 

1.  if  by  6.  2.  ftf  by  5.  6.  ^  by  12. 

Process.  3.  ft  by  8.  7.  &  by  15. 

/,       ,        2         4.  W-by6.  8.  «fby5. 

is  X  ?  =  Is        5.  af.  by  60.  9.  M*  by  9. 

6.  What  is  the  value  of: 

i-  H-8-*?   4-  *V-¥?      7.  ft-*? 

2-  If  -*-  1  ?     5.  if  -.-  1  ?  8.  ff  -  il  ? 

_3-  H-i-}?     6.  If-l?  9.  M-H? 

7.  Find  the  value  of: 

1.  f  of  |  of  16  -s-  f  of  f  of  5J, 


Process. 


3__25__-,9 

"  16  ~       16* 


114  PRACTICAL    ARITHMETIC 

2.  £  of  f£  of  51  -h  41  times  l  of  17, 

3.  I  of  21-51 

4.  l  of  |  of  ^  by  7  times  |  of  f 

5-  (^  +  T3o)  X  TV 

6.  I  of  41  -  |  of  3f . 

7.  f  of  f  of  15  -h  |  of  |  of  6. 

8.  21  of  21  -  ^  of  3f . 

9.  f  of  -fr  ot  22  -f-  T%  of  f  Of  i6. 

10.  |  of  -f-  of  if  -  6. 

11.  ^  of  3|  of  6  -f-  -J  of  6  times  If 

12.  fof  3iof^-5i 

13.  81  times  £  of  7  -=-  f  of  f  of  5. 

14.  ^  -  I  of  21  of  If 

15.  I  of  351  -4-  f  of  8|. 

16.  Aoflf-fof|. 

17-  A  of  ir  -*-  A  of  II- 

18.  Wofi|-2Vof3^. 

19.  |offof|-|ofi|of  ff. 

20.  Joffof|-|ofTVofiloflf. 

PROBLEMS. 

1.  If  7f  yards  of  cloth  cost  $47^,  what  is  the  price  per 
yard  ? 

2.  If  a  man  spends  $f  per  day  for  cigars,  in  how  many 
days  will  he  spend  $17-^? 

3.  If  -§-  of  a  ton  of  hay  costs  $15,  what  is  the  cost  of  one 
ton? 

4.  A  man  has  229^-  pounds  of  honey,  which  he  wishes  to 
pack  in  boxes  containing  8^-  pounds  each.     How  many  boxes 
will  he  require  ? 

5.  A  man  owning  ^|-  of  a  ship,  sold  -|  of  his  share,  and 
divided  the  remainder  equally  among  his  three  sons.     What 
part  of  the  ship  did  each  son  own  ? 


COMPLEX  FRACTIONS  115 

6.  The  product  of  two  numbers  is  ^f,  and  one  of  the 
numbers  is  1^.     What  is  the  other  number? 

7.  What  number  multiplied  by  1-|  will  produce  14^? 

8.  How  many  yards  of  cloth  at  $3f  per  yard  can  be 
bought  for  $317f? 

9.  When  wheat  is  selling  at  $1-J  per  bushel,  how  many 
bushels  can  be  bought  for  $3168? 

10.  For  $8^  how  many  thousand  feet  of  gas  at  $1J  per 
thousand  can  be  bought  ? 


COMPLEX  FRACTIONS. 
A  Complex  Fraction  has  a  fraction  in  one  or  both  of  its 


1.  Simplify  S. 

Process.  Explanation. 

^  -•     6  4  =  :  "¥•     (  A  ^  H  Since  ||  signifies  that  5^  is  to  be 

divided  by  6£,  we  proceed  according  to  the  rule  for  division,  and  obtain  ft. 

2.  Simplify  : 


116  PRACTICAL   ARITHMETIC 

16. 


' 


n       /\    o  4.' 

A     3— i 

T  ~~~  T!"  ^"  9  Q 

FRACTIONAL  RELATIONS. 

1.  In  the  equation,  -J-  of  4  =  2,  the  ^  expresses  the  relation 
of  2  to  4.     If  the  question  is  asked,  "  What  is  the  fractional 
relation  of  2  to  4  ?"  the  answer  simply  reverses  the  equation, 
— "  2  =  1  of  4." 

This  equation  may  be  derived  analytically,  thus :  Since  1 
=  J-  of  4,  2,  being  twice  1,  =  f  or  J  of  4. 

2.  In  like  manner  show  the  fractional  relation  of  3  to  9. 
Of  5  to  8. 

What  part  of  8  is  5  ?     Does  the  answer  show  the  relation 
of  5  to  8? 

3.  What  part  of  $5  is  $1? 

Since  $1  ==  i-  of  $5,  $1,  being  1  of  $1,  =  £  of  £  of  five 
dollars,  or  ^  of  five  dollars. 

4.  In  like  manner  find  the  fractional  relation  of  $f  to  $6. 

5.  What  part  of  7  acres  is  |-  of  an  acre? 

6.  Is  $5  any  part  of  10  acres? 

7.  What  is  the  fractional  relation  of  7  men  to  9  trees  ? 

PRINCIPLE. 

Only  like  numbers  can  have  fractional  relation  to  each 
other. 


FRACTIONAL  RELATIONS  117 

EXERCISES. 
To  find  the  Fractional  Relation  between  Two  Numbers. 

1.  Form  an  equation  to  show  the  fractional  relation  of: 

1.  8  to  24.  11.  |  to  5.  21.  6f  to  40. 

2.  13  to  26.  12.  f  to  10.  22.  6J  to  425. 

3.  12  to  18.  13.  f  to  7.  23.  2£  to  42. 

4.  10  to  15.  14.  -f  to  9.  24.  6£  to  128. 

5.  9  to  27.  15.  f  to  16.  25.  6f  to  75. 

6.  35  to  40.  16.  f  to  26.  26.  12^-  to  180. 

7.  16  to  24.  17.  |  to  7.  27.  1  of  3|  to  84. 

8.  15  to  35.  18.  f  to  15.  28.  f  of  f  to  75. 

9.  19  to  95.  19.  |  to  16.  29.  8^  to  ^  of  90. 
10.  20  to  110.  20.  -&  to  3.  30.  J  of  2|  to  £  of  18. 

2.  Find  the  fractional  relation  of: 

1.  ftof. 

Suggestion  :  f  ==  j£  and  f  =  &  ;  10  =  ty  of  9. 

2.  $i  to  $£.  12.  2J-  to  f . 

3.  $f  to  $f .  13.  1£  to  2|. 

4.  1^  to  $£.  14.  2|  to  7|. 

5.  $|  to  $1.  15.  74-  to  2|. 

6.  ^tofrfr.  16.  3Jto8f 

7.  $f  to  $|.  17.  $6  to  $100. 

8.  $f  to  $£  18.  $8f  to  $100f . 

9.  $f  to  $f  19.  f  of  If  to  3£. 

10.  $f  to  $f  20.  9^P-  to  12^-. 

11.  $T3g-  to  $f.  21.  If  to  31  X  f  of  f 

To  find  a  Number  from  its  Fractional  Relation  to  Another 

Number. 

1.  Tn  the  equation,  3  =  -^  of  9,  the  ^  expresses  the  relation 
of  3  to  9. 


118  PRACTICAL   ARITHMETIC 

2.  If  the  question  arise,  3  is  ^  of  what  number  ?  what  is 
the  answer? 

3.  The  analytical  answer  is  what? 

Suggestion  :  %  of  the  number  —  3  ;  f,  the  whole  of  the  number,  =  what  ? 

4.  Answer  the  following  inquiries  : 

1.  12  is  ^  of  what  number? 

2.  24  is  £  of  what  number  ? 

Suggestion :  £  of  the  number  =  ^  of  24. 

3.  24  is  -|  of  what  number  ? 

4.  28  is  f-  of  what  number? 

5.  48  is  %  of  what  number  ? 

6.  TT-  is  f  of  what  number  ? 

Suggestion  :  ^  of  the  number  =  ^  of  ffi. 

7.  -Jf  is  -^  of  what  number? 

8.  !~|  is  if-  of  what  number? 

9.  ^2.  is  HI  of  what  number? 
10.  l  of  3£  is  -f-  of  what  number? 

REVIEW. 

1.  The  sum  of  two  fractions  is  f ,  and  their  difference  is  \. 
Required  the  fractions. 

Suggestion  :    Were  the  difference  0,  the  fractions  would  be  |  and  ^. 
Hence  the  greater  fraction  =  f  -f  i  of  \ ;  the  less  =  f  —  |  of  | . 

2.  If  a  man  can  cut  in  one  day  ^  of  a  field  containing  7 
acres  of  wheat,  how  many  acres  can  he  cut  in  ^  of  a  day  ? 

3.  Reduce  -J-,  -| ,  f  and  ^  to  equivalent  fractions  whose 
denominators  shall  be  24. 

4.  Add  ^,  ff,  4f,  15|,  and  explain  fully. 

,  4  of  4  of  7f 

5.  Find  the  value  of  a — - &. 


FRACTIONAL  RELATIONS  H9 

6.  The  product  of  three  numbers  is  % ;  two  of  the  num- 
bers are  2-J-  and  %  •  what  is  the  third  ? 

7.  A  housekeeper  bought  6  mahogany  chairs  at  3|-  dollars 
each,  and  gave  for  them  2  ten-dollar  bills  and  one  five-dollar 
bill.     What  change  ought  she  to  receive? 

8.  Find  the  sum  of  1^-  +  3J4-  -f  4lf. 

9.  A  box  contains  345  eggs.     What  is  their  value  at 
$.16f  a  dozen? 

10.  If  |-  of  an  acre  of  land  cost  101  dollars,  what  will  f 
of  an  acre  cost? 

11.  Reduce  816T5T  to  an  improper  fraction. 

12.  Subtract  l  of  T9¥  from  8^  "^  24\ 

16  1         21 

13.  Simplify  ^  ;  also,  p;  —  y • 

14.  If  T3jj-  of  an  acre  of  land  is  worth  $79^,  what  is  1  acre 
worth  ? 

15.  Reduce  ^ — 1  to  a  simple  fraction. 

«i 

16.  From  what  must  6f  be  subtracted  to  leave  1  of  3|? 

17.  At  -2-  of  a  dollar  per  bushel,  what  will  be  the  cost  of  £ 
of  a  bushel  of  potatoes? 

18.  |  of  27  is  f  of  what  number? 

19.  From  f  of  f  take  £  of  |. 

20.  Simplify  ^_X  18  X  75  X  6^ 

7  25  X  17  X  14  X  9 

21.  A  merchant  paid   85^  dollars   for   15^-  tons  of  coal. 
How  much  did  the  coal  cost  him  per  ton  ? 

22.  Simplify  8^  -  2£  -  3J  +  6^  -  5|. 

23.  150  is  ^  of  what  number? 

24.  $150  was  paid  for  a  horse  and  saddle.     If  the  cost  of 
the  saddle  was  ^  of  the  cost  of  the  horse,  what  was  the  cost 
of  each  ? 


120  PRACTICAL  ARITHMETIC 


25. 

26.  Find  the  quotient  of  20f  -t-  -ft  of 

27.  How  many  lemons,  at  -f^  of  a  dollar  a  dozen,  will  pay 
for  80  oranges  at  2-J-  cents  each  ? 

28.  Four  loads  of  hay  weigh  respectively  1723f,  231  7f, 
1547f,  and  1357^  pounds.     What  is  the  total  weight  of  the 
hay? 

29.  Reduce  ifff  to  its  lowest  terms. 

30.  A  farmer  had  ^  of  his  sheep  in   one  pasture,  \  in 
another,  and  the  remainder,  which  were  77,  in  a  third  pasture. 
How  many  sheep  had  he? 

31.  If  I  give  A.  \  of  my  money,  B.  ^  of  it,  and  C.  \  of 
it,  what  part  of  my  money  have  I  left  ? 

32.  The  divisor  is  -ffg,  and  quotient  -f-|-|.     What  is  the 
dividend  ? 

33.  A  man  bought  land  for  $5130,  and  sold  it  so  as  to 
gain  y^  of  the  cost,  the  gain  being  $3  per  acre.     How  many 
acres  did  he  buy  ? 

34.  After  buying  a  suit  of  clothes  for  $60  I  found  I  had 
^  of  my  money  left.     How  much  had  I  at  first  ? 

35.  What  number,  diminished  by  the  difference  between 
J  and  £  of  itself,  leaves  a  remainder  of  34  ? 

36.  Divide  f  of  3£  by  f  of  {%. 

37.  If  3J  bushels  of  oats  will   sow  an   acre,  how  many 
bushels  will  it  take  to  sow  7-^  acres? 

38.  Reduce  *«**  X       * 

2*  i  of  i 

39.  Find  the  value  of  5  +  6f  —  7T7F  +  ||. 

40.  The  circumference  of  a  bicycle  wheel  is  7^  feet  ;  the 
circumference   of  another   bicycle   wheel    is    7^   feet.     How 
many  more  times  will  the  smaller  wheel  turn  than  the  larger 
ingoing  5280  feet? 

41.  Divide  (|  -  |)  by  (f-  |). 


FRACTIONAL  RELATIONS  121 

42.  Reduce  ^  to  786ths. 

43.  If  -f-  of  a  cord  of  wood  cost  $6|-,  what  will  10  cords 
cost? 

44.  Find  the  sum  and  product  of  |-,  -J  and  -f . 

45.  What  is  the  value  of  (f  of  f  of  3f  +  8|)  -^  (10£  - 

7  A)? 

46.  A  shepherd,   being  asked  how   many  sheep  he   had, 

answered  that  -f-  of  f  of  the  whole  number  was  45.     How 
many  had  he? 

47.  Find  the  G.  C.  D.  and  the  L.  C.  Dd.  of  833,  1127, 
1421,  343. 

48.  What  part  of  f  is  £? 

49.  If  3|-  yards  of  cloth  cost  84  cents,  how  much  is  that 
per  yard  ? 

50.  The  reciprocal  of  -|  is  1  -f-  -J.     What  are  the  sum,  the 
difference,  and  the  product  of  -J  and  its  reciprocal  ? 

51.  Reduce  to  a  common  denominator  and  add  -|  X  £  X  |-j 

A,  I.  and  -ft- 

52.  A  lot  which  cost  $400  was  sold  for  $480.     What  part 
of  the  cost  was  gained  ? 

53.  How  much  is  the  sum  of  ^,  |-,  ^  greater  or  less  than  | 
of  the  sum  of  11  1? 

54.  If  bricks  cost  $8.50  a  thousand,  what  is  the  cost  of 
one  brick? 

55.  Simplify  -jf-«     Give  the  principles  involved. 

12 

56.  A  regiment  lost  in  battle  250  men,  which  was  f  of 
the   regiment.     What  was   the   number  of  men    before   the 
battle  ? 

57.  Divide  98  by  11-^,  and  multiply  the  quotient  by  f 
of  8f. 

58.  Reduce  T7F,  ii,  fj  to  their  L.  C.  D. 


122  PRACTICAL  ARITHMETIC 

59.  A  barrel  of  beef,  which  holds  200  pounds,  was  |-  full. 
How  many  pounds  would  there  be  left  in  it  after  53f  pounds 
were  taken  out  ? 

60.  A  man  having  $5^  bought  a  knife,  and  then  had  left 
$4T9g-.     How  much  did  the  knife  cost? 

61.  Mr.  Gould  sold  a  cow  for  $30,  which  was  f  of  what 
she  cost  him.     How  much  did  he  lose? 

62.  At  9^  dollars  a  barrel,  how  many  pounds  of  flour  can 
be  bought  for  $3£?     [One  barrel  =196  pounds.] 

63.  When  hay  is  worth  $9J  a  ton,  what  will  f  of  3|-  tons 
cost? 

64.  A.  and  B.  kill  an  ox.     A.  takes  f  and  B.  the  re- 
mainder.    If  B.'s  share  weighs  361^  pounds,   what  is  the 
weight  of  the  ox  ? 

65.  What  fraction  of  18f  is  6f  ? 

66.  If  2f  acres  of  land  cost  $220,  what  will  be  the  cost  of 
17-J  acres?     Indicate  the  work  and  cancel. 

67.  If  15  men  do  a  piece  of  work  in  10-|  days,  how  long 
would  it  take  5  men  to  do  the  same  work  ? 

68.  If  5  be  added  to  both  terms  of  the  fraction  |-,  will  its 
value  be  increased  or  diminished  ? 

69.  If  A.  can  do  a  piece  of  work  in  5  days  and  B.  in  8 
days,  how  long  will  it  take  both  to  do  it  ? 

70.  There  are  two  numbers  whose  sum  is  140,  one  of  which 
is  f  the  other.     What  are  the  numbers  ? 

71.  The  product  of  two  numbers  is  6,  and  one  of  them  is 
1846.     What  is  the  other? 

72.  If  3  dozen  lemons  cost  $1|,  what  will  be  the  cost  of 
56  lemons? 

73.  If  7^-  pounds  of  rice  cost  $.90,  how  many  pounds  can 
be  bought  for  $1.10? 

74.  A  clerk  earns  $lf  a  day,  and  spends  $-§-  a  day.     How 
much  does  he  save  in  a  year? 


FRACTIONAL  RELATIONS  123 


75.  Multiply  by 

TO.  (f|  x  ff)  -5-  («  x  f|)- 

77.  Margaret,  in  attempting  to  divide  a  fraction  by  |~|,  in- 
verted the  dividend   instead  of  the  divisor,  and  obtained   a 
quotient  of  ^J-.     What  was  the  given  fraction  ? 

78.  If  a  man's  brain  is  ^  of  his  weight,  and  weighs  3-^ 
pounds,  what  is  his  weight? 

79.  Which  is  the  greater,  -fifa  or  -^  ? 

80.  When  land  is  worth  1  00  dollars  per  acre,  what  part  of 
an  acre  will  be  worth  26f  dollars  ? 

81.  A  cistern  can  be  filled  by  one  pipe  in  15  hours,  and  by 
another  in  20  hours.     In  what  time  can  the  two  pipes  fill  it 
flowiug  together  ? 


82.  What  part  of        "  is  ? 

t  i 

83.  What   is   the   quotient  of   1^   divided   by    its   recip- 

rocal? 

NOTE.  —  The  reciprocal  of  a  quantity  is  1  divided  by  that  quantity. 

84.  Change  f  to  a  fraction  whose  denominator  shall  be 
35. 

85.  Find  the  least  number  of  apples  that,  arranged   in 
groups  of  8,  9,  10,  or  12,  will  have  just  6  over  in  each  case. 

86.  Three  times  a  number  plus  -|  of  it,  plus  4f  times  the 
number  plus  ^  of  it,  are  how  many  times  the  number  ? 

87.  If  f  of  a  steeple  casts  a  shadow  83-|  feet  long,  how 
long  is  the  shadow  cast  by  f  of  it? 

88.  A  man  has  4^  bushels  of  potatoes,  which  is  -|  of  the 
quantity  that  he  planted.     How  many  did  he  plant  ? 

89.  A  man  who  received  -J-  of  his  father's  property  gives 
to  his  own  son  J  of  what  he  received.     Who  then  has  -fa  of 
the  whole  ? 


124  PRACTICAL   ARITHMETIC 

90.  Two  men  require  8^  days  to  take  account  of  a  stock 
of  goods.     Six  men  would  need  what  time  ? 

91.  What  fraction  is  the  quotient  of  -fffc  -=-  ^? 

92.  In  sowing  a  field,  one  kind  of  seed  is  used  at  the  rate 
of  12^  bushels  to  5  acres.    What  will  be  required  to  sow  22f 
acres,  using  -f-  as  much  to  the  acre  as  before  ? 

93.  When  oysters  yield   1J  gallons  to  the  bushel,  a  25- 
gallon  barrel  can  be  filled  from  how  many  bushels  in  the 
shell? 

94.  l  of  a  bushel  of  berries  is  picked ;  ^  of  them  are  sold 
to  one  man,  1  of  the  remainder  to  another.     What  fractional 
part  remains  unsold  ? 

95.  Oranges  are  bought  at  3  for  $.05  and  sold  at  4  for  $.09. 
What  is  gained  on  a  box  of  9  dozen,  1  in  12  of  which  are 
worthless. 

96.  21  —  f  —  |  —  |  —  |  =  ? 

97.  Find  the  G.  C.  D.  and  the  L.  C.  Dd.  of  45,  90,  100, 
and  200. 

98.  A.,  B.,  and  C.  can  do  a  piece  of  work  in  10  days.     A. 
can  do  it  in  25  days,  and  B.  in  30  days.     In  what  time  can 
C.  doit? 

99.  A.  and  B.  together  had  $5700.     f  of  A.'s  money  was 
equal  to  -J  of  B.'s.     How  much  had  each  ? 

100.  Define: 

1.  Fraction.  10.  Reduction. 

2.  Decimal  fraction.  11.  Higher  terms. 

3.  Common  fraction.  12.  Lower  terms. 

4.  Fractional  unit.  13.  Lowest  terms. 

5.  Denominator.  14.  Like  fractions. 

6.  Numerator.  15.  Unlike  fractions. 

7.  Proper  fractions.  16.  Common  denominator. 

8.  Improper  fractions.  17.  Least  common  denominator. 

9.  Mixed  number.  18.  Fractional  relation. 


DECIMAL  FRACTIONS  125 

101.  Kepeat: 

1.  The  principles  of  Addition  of  Fractions. 

2.  The  rule  for  Addition  of  Fractions. 

3.  The  brief  directions  for  finding  L.  C.  D.  of  Fractions. 

4.  The  principles  of  Subtraction  of  Fractions. 

5.  The  rule  for  Subtraction  of  Fractions. 

6.  The  principles  of  Multiplication  of  Fractions. 

7.  The  rules  for  Multiplication  of  Fractions. 

8.  The  principles  of  Division  of  Fractions. 

9.  The  rules  for  Division  of  Fractions. 
10.  The  principle  of  Fractional  Eelation. 

102.  Invent  and  solve  : 

1.  Five  problems  in  Reduction  of  Fractions. 

2.  Five  problems  in  Addition  of  Fractions. 

3.  Five  problems  in  Subtraction  of  Fractions. 

4.  Five  problems  in  Multiplication  of  Fractions. 

5.  Five  problems  in  Division  of  Fractions. 

6.  Five  problems  in  Relation  of  Fractions. 

7.  Five  miscellaneous  problems  in  Fractions. 


DECIMAL  FRACTIONS. 

DEFINITIONS. 

1.  A  Decimal  Fraction  denotes  one  or  more  of  the  decimal 
divisions  of  a  unit. 

2.  Decimal  Fractions  are  usually  called  decimals  (Latin, 
decem,  "ten"). 

3.  A  Pure  Decimal  consists  of  decimal  figures  only,  as 
.234. 


126  PRACTICAL   ARITHMETIC 

4.  A  Mixed  Decimal  consists  of  an  integer  and  a  decimal, 
as  23.005. 

5.  A  Complex  Decimal  has  a  common  fraction  on  the 
right  of  the  decimal,  as  .06f . 

NOTATION  AND  NUMERATION. 

1.  By  placing  a  mark  (.),  called  the  decimal  point,  after 
units  of  the  first  order,  the  numeration  and  notation  table  is 
extended  to  express  parts  of  a  unit  on  the  decimal  scale. 

2.  The  relation  of  decimals  and  integers  to  each  other  is 
clearly  shown  by  the  following 

Numeration  Table. 


«  .      H  3 

I  I        ,  *        |  ,;        § 


s 

| 

•s 

^H 

1 

^ 

N 

6 

CO 

.2 

g 

1 

1 

1 

1 

1    1 

1 

s 

CO 

o> 

1 

P 

O 

1 

Is 

£ 

1 

G 

w 

G 

o> 

H 

1 

G 

w 

G       -1 

H     ^ 

c 

0 

C/3 

c 
0 
H 

s 
P 

1 

c 

w 

1 

G 

o> 
H 

c 

w 

s 

1 

C 

W 

9 

8 

7 

6 

5      4 

3 

2 

i  . 

2 

3 

4 

5 

6 

7 

8 

9 

INTEGERS.  DECIMALS. 

By  examining  this  table  we  see  that :  — 

Tenths  are  expressed  by  one  figure. 

Hundredths  are  expressed  by  two  figures. 

Thousandths  are  expressed  by  three  figures. 

Ten  thousandths  are  expressed  by  four  figures. 

Hundred  thousandths  are  expressed  by  five  figures. 

Millionths  are  expressed  by  six  figures. 
3.  The  decimal  point  is  a  separatrix,  not  a  period ;  it  is 
read  "and." 


DECIMAL  FRACTIONS  127 

Remember  that  the  name  of  the  6th  decimal  order  is 
Millionths,  and  give  orally  the  names  of  the  following  orders  : 
6th  order,  5th  order,  4th  order,  3d  order,  2d  order,  1st  order, 
3d  order,  5th  order,  4th  order,  6th  order,  1st  order,  5th  order, 
2d  order,  4th  order,  3d  order,  6th  order,  5th  order,  4th  order, 
3d  order,  2d  order,  1st  order,  6th  order. 

In  what  decimal  place  do  you  find :  Millionths  ?  Thou- 
sandths? Tenths?  Hundredths?  Ten-thousandths ?  Hun- 
dred-thousandths? Ten-millionths  ?  Hundredths?  Mil- 
lionths ?  Thousandths  ? 

4.  Eead  the  following:  1.2,  1.03,  1.004,  1.0005,  1.00006, 
1.000007,  2.008,  3.09,  4.0001,  5.000002,  6.00003,  7.0004, 
8.9,  9.10. 

PRINCIPLES. 

1.  Decimals  and  integers  are  subject  to  the  same  law 
of  local  value. 

2.  Each  cipher  inserted  between  the  decimal  point  and 
the  first  figure  of  a  decimal  diminishes  the  value  of  the 
decimal  ten-fold. 

3.  Annexing  ciphers  to  a  decimal  does  not  alter  its  value. 

.05  =  .050,  for  0  thousandths  add  nothing  to  5  hundredths. 

4.  The  denominator  of  a  decimal,  when  expressed,  is  1 
with  as  many  ciphers   annexed  as   there  are   orders,  or 
places,  in  the  decimal. 

Read  7.039. 

ANALYSIS. 

7  is  an  integer  representing  7  units,  and  is  read  "  seven."  The  decimal 
point  is  read  "and  "  0  denotes  the  absence  of  tenths,  and  is  not  read. 
3  hundredths  +  9  thousandths  is  read  "  39  thousandths."  Hence  7.039  is 
read  "  7  and  39  thousandths  " 

RULE. 

Bead  the  decimal  as  an  integral  number,  and  add  the 
decimal  name  of  the  right-hand  figure. 


128 


PRACTICAL  ARITHMETIC 


EXERCISES. 

1.  Read  the  following  : 

1.  .7. 

2.  .36. 

3.  .625. 

4.  .025. 

5.  .0005. 

6.  .12345. 

7.  .789123. 

8.  .405607. 

9.  .890123. 

10.  .456789. 

11.  8.54. 

12.  85.4. 

13.  9.213. 

14.  7.389. 

15.  12.3601. 

16.  19.0032. 

17.  25.00081. 

2.  Write  decimally  13  thousandths. 


18.  29.15625. 

35.  6.839. 

19.  341.63456. 
20.  1001.000089. 
21.  .6305. 

36.  .24|. 
37.  3.70& 

38.  7.039. 

22.  .446|. 
23.  .00371. 

39.  8.1367. 
40.  7.0308f. 

24.  .0506. 
25.  .087345.   • 

41.  9.1007& 
42.  146.0302056. 

26.  6.00056. 

43.  376.932474. 

27.  11.04735. 

44.  2.234006. 

28.  63.04048. 

45.  487.000081035. 

29.  100.000001. 

46.  586.0004003256, 

30.  734.819181. 

47,  .5. 

31.  341.63456. 

48.  5.078. 

32.  .684. 

49.  8.008. 

33.  .084. 

50.  6.2040. 

34.  .004. 

51.  37.40253. 

ANALYSIS. 

13  thousandths  =  one  hundredth  -f-  3  thousandths.  0  tenths  are  given. 
As  the  number  is  a  pure  decimal,  the  expression  of  it  must  begin  with  the 
decimal  point.  Hence  13  thousandths  expressed  decimally  is  .013. 

RULE. 

"Write  the  number  as  an  integer,  and  give  the  right-hand 
figure  the  place  indicated  by  the  decimal  name  of  the 
number. 


3.  Express  decimally : 

1.  Seven  tenths. 

2.  Nine  tenths. 


Twelve  hundredths. 
Seventeen  hundredths. 


DECIMAL  FRACTIONS  129 

3.  Four  hundredths.     42  hundredths. 

4.  125  thousandths.     22  thousandths. 

5.  20  hundredths.     Eight  thousandths. 

6.  30  thousandths.     206  thousandths. 

7.  3027  ten-thousandths. 

8.  Three  hundred  ten-thousandths. 

9.  Forty-two  ten-thousandths. 

10.  145  hundred-thousandths. 

11.  Fifty-one  hundred-thousandths. 

12.  One  hundred  seven  million  ths. 

13.  306  ten-millionths. 

14.  3259  hundred-thousandths. 

15.  429  ten-millionths. 

16.  4268  hundred  millionths.. 

17.  13,760  millionths. 

18.  Three  hundred  forty-two  millionths. 

19.  One  hundred  forty-five  hundred  thousandths. 

20.  703,205  millionths. 

4.  Express  as  mixed  decimals  the  following  : 

1.  5&.  8-  24^V  15. 

2-  7JV.  9.  27^.  16. 

3.  8T^.  10.  54^^.  17. 

4.  ftrfo.  11.  74T¥|W.  18. 

5.  12^.  12.  M&M,.  19. 

6.  16^Hhr.  13.  48.  20. 
t.  I^^^'           ^* 


UNITED  STATES  MONEY. 

1.  Read  $12.925  as  a  mixed  decimal,  and  as  dollars,  cents, 
and  mills. 

It  is  read  u  12  and  ^-thousandths  dollars,"  or  "12  dollars, 
ninety-two  cents,  five  mills." 


130  PRACTICAL    ARITHMETIC 

2.  Read  in  like  manner  the  following  : 

1.  $89.06.  5.  $59.375.  9.  $1.375. 

2.  $94.254.          6.  $86.047.  10.  $0.876. 

3.  $69.045.          7.  $344.002.  11.  $0.093. 

4.  $195,005.        8.  $20.25.  12.  $0.001. 

3.  Express   decimally    $T4¥8o,    $20J,    $35^    $rjfo,    $4^, 


five  cents,  five  dimes,  five  mills,  five  dollars  five  cents  five 
mills. 

REDUCTION. 
To  Like  Decimals. 

$T£  T  =  $^0-     Therefore  $.06  =  $.060. 


PRINCIPLE. 
Annexing  ciphers  to  a  decimal  does  not  alter  its  value. 

EXERCISES. 

1.  Reduce  .7,  .05,  and  .304  to  like  fractions. 

Process.  Explanation. 

^  —  :  .700  Thousandths  is  the  lowest  order  given,  hence  all 

QK  _    Q-Q  the  fractions  must  be  reduced  to  thousandths.     Since 

annexing  ciphers  to  a  decimal  does  not  alter  its  value, 
-  .olH  we  annex  two  ciphers  to  .7,  thus  rendering  it  700 

thousandths  ;  one  cipher  to  .05,  thus  rendering  it  50  thousandths. 

RULE. 

By  annexing  ciphers  give  each  decimal  the  same  number 
of  decimal  places. 

2.  Reduce  to  like  decimals  the  following  : 

1.  .25,  .025,  .37. 

2.  .523,  4.36,  5.0315. 

3.  .4036,  .5,  .375. 


DECIMAL  FRACTIONS  131 

4.  .06,  .008,  .4267,  .026. 

5.  .409,  3.61,  .75,  .10055,  19.6. 

6.  7.07,  5.0909,  1.9090,  19.099. 

7.  .12,  .8,  306.973,  .004,  48.56. 

8.  .0436,  .04506,  .82. 

9.  .8104,  .0008,  8000.4. 
10.  8.1,  .43,  .68,  3.96. 

To  a  Common  Fraction. 

1.  What  is  the  denominator  of  .125  ? 

2.  What  is  its  numerator  ? 

3.  Write  .125  as  a  common  fraction. 

4.  What  part  of  the  expression  .125  did  you  omit? 

EXERCISES. 
1.  Reduce  .375  to  a  common  fraction. 

Process. 


RULE. 

"Write  the  decimal,  omitting  the  decimal  point;  supply 
the  decimal  denominator,  and  reduce  the  fraction  to  its 
lowest  terms. 

2.  Reduce  the  following  decimals  according  to  the  rule  : 

1.  .45.  8.  4.0125.  15.  23.075. 

2.  .027.  9.  .4355.  16.  .354. 

3.  .72.  10.  10.25.  17.  .00625. 

4.  1.39.  11.  .0005.  18.  .05375. 

5.  .375.  12.  .5000.  19.  15.064. 

6.  .625.  13.  10.25.  20.  .005396. 

7.  4.75.  14.  15.725.  21.  .0007890. 


132  PKACTICAL   ARITHMETIC 

COMPLEX    DECIMALS. 
EXERCISES. 

1.  Reduce  .9^  to  a  common  fraction. 

Process.  Explanation. 

,9£  =  |i-  =  ff  =  If  Multiplying  both  terms  of  *$  by  3,  we 

obtain  f  $. 

2.  Reduce  in  like  manner : 

1.  .16f.  8.  .04|.  15.  $66.66f. 

2.  .3}.  9.  .0371  16.  $25.14f 

3.  .561  10.  .5621  17.  $50.061 

4.  .33t.  11.  $5.9f  18.  $100.871 

5.  .121  12.  $12.18£.  19.  $700.371 

6.  .16f.  13.  $33.031  20.  $1000.111 

7.  .871  14.  J55.83J.  21.  $33.621. 

COMMON    FRACTIONS. 

1.  What  is  the  denominator  of  a  common  fraction  that 
may  be  directly  expressed  as  a  decimal? 

2.  If  ^  be  reduced  to  a  decimal,  what   is   the   smallest 
denominator  it  can  have? 

3.  |  =  how  many  lOths? 

4.  How  is  ^5-  written  decimally  ?     How  -f  ? 

2  units  _  20  tenths  _  4 

5                   5 
5  units 5000  thousandths ^5 

8  8 

5.  How  does  the  number  of  places  in  the  quotients  agree 
with  the  number  of  ciphers  annexed  ? 


DECIMAL  FRACTIONS  133 

EXERCISES. 

1.  Reduce  f  to  a  decimal. 

Process.  Explanation. 

JLJLQJL  —  .375  We  find  by  trial  that  three  ciphers  must  be  an- 

nexed to  3  to  secure  a  complete  quotient.     The  three 

ciphers  annexed  require  the  pointing  off  of  three  decimal  places  in  the 
quotient. 

RULE. 

Annex  ciphers  to  the  numerator  and  divide  by  the 
denominator.  Point  off  in  the  quotient  as  many  decimal 
places  as  there  are  ciphers  annexed. 

2.  Reduce  the  following  to  decimals  : 

1.  i.  9.  |if.  17.  Jf  25. 

2.  f,  10.  f.  18.  |f.  26. 

3.  f.  11.  f.  19.  |f.  27. 

4.  f  12.  f.  20.  ff  28. 

^5  1Q        1  91       51  9Q 

o.  yj.  M.  YJ-.  zi.  fT.  z».  -nnnnr- 

6.  A-  14.  A-  22.  fft.  30. 

7.  if.  15.  if  23.  tfff.  31. 

8    1 7.  16  * 3  24      7  32 

NOTE. — It  is  not  possible  in  every  case  to  render  the  division  exact  by 
annexing  ciphers.  Frequently  a  remainder  occurs,  which  may  be  used  as 
the  numerator  of  a  fraction  ;  or  it  may  be  disregarded,  and  the  sign  -(- 
employed  to  denote  the  incompleteness. 

3.  Reduce  f  to  a  decimal. 

Process. 

JLJLO_Q_<L  --  .4285f  or  .4285  +. 

4.  Reduce  to  decimals  the  following  : 

16.  -j^. 

17. 

18. 

19.  TW^. 

20. 


1.  f 

6-  A- 

11-11 

2.  f 

7-  A- 

12-  f 

3-  A- 

8.  ^j-. 

13.  -H 

A        7 

9.  |f. 

14.  fJ 

5'.  f 

10-  A. 

15.  f| 

134  PRACTICAL   ARITHMETIC 

5.  Give  decimal  form  to  the  fractions  in  these : 

1.  16f  6.  76£.  11.  31.0f 

2.  35£.  7.  981  12.  .000^. 

3.  .93f  8.  .54}.  13.  3.00^. 

4.  4.5f  9.  5.32f  14.  4627^. 

5.  .34f.  10.  48.6f.  15.  1899£f 

ADDITION. 

INDUCTIVE   STEPS. 

*•  A  +  A  =  what?  .7  +  .2  =  what?  .7  +  .02  = 
what?  .7  +  .32  =  what? 

2.  If  .7  +  .02  =  .72,  and  if  .7  +  .32  =  1.02,  does  the 
addition  of  decimals  differ  from  the  addition  of  integers  ? 

Obviously,  the  addition  of  decimals  and  the  addition  of 
integers  are  governed  by  the  same  principles. 

EXERCISES. 
1.  What  is  the  sum  of  4.18,  .005,  and  5.6057? 

Process.  Explanation. 

4. IS        —  4.1800  By  writing  the  decimal  points  in  a  vertical 

AAK  r\r\ZK.  column  the  dibits  of  like  orders  fall  in  the  same 

.UUD     =     .UUOO  e  .  .  . 

.^ -7  column,  and  we  proceed  as  in  addition  of  inte- 

5.6057  =  5.6057         gers     To  mark  the  place  of  tenths  in  the  gum 

9.79 1  2  we  put  a  decimal  point  therein  under  tho  column 

of  points. 

RULE. 

Write  the  decimals  so  that  the  decimal  points  will  fall 
in  a  vertical  column;  add  as  in  addition  of  integers,  and 
place  a  point  in  the  sum  under  the  column  of  points. 


ADDITION  OF  DECIMAL  F&ACTIONS  135 

2.  Find  the  value  of: 

1.  87.79  -h  47.05  -f  245.406. 

2.  .027  +  1.39  +  48.6  -f  72.978. 

3.  4  dollars  +  16  cents  +  87  cents  +  95  mills. 

4.  $.047  +  $6.210  +  $.47  -f  3  mills. 

5.  12.34  -f  432.015  +  302.23  +  .00025. 

6.  106  +  .106  +  1.06  +  10.6. 

7.  $5.18  +  $3.09  +  $46  -f  $54.185. 

8.  25.002  -f-  206.31  +  505.05  +  1.015  +  8.33. 

9.  .718  +  643.5  +  29.21  -f  114. 

10.  72.5  -f  7.29  -f  4.009  +  3.275  +  .4. 

11.  .9374  +  13.21  +  45.135  +  1.0006  -f  82.16. 

12.  9  tenths  +  18  hundredths  +  125  thousandths  + 

125. 

13.  .61692  -f  243.734  -f  901  +  68.45213  +  8.386. 

14.  2.352  +  .0008  +  5.0856  +  9.6823. 

15.  $63.75  -f  $9.60  +  7-1-  cents  -f  $.80  -f  $|  +  $.375. 

16.  $364.12|  +  $1.18}  +  $100.25  +  $63.50. 

17.  $48TV5o  +  $-97  -f  ($f  +  $.62-1  +  $|). 

18.  .3  -f  2.03  +  ft  -f  m  +  ftfr  +  6.4837. 

19.  .1  +  .01  +  .001  +  .0001  +  .00001  -f  11.111. 

20.  $9.90  +  $99.99  -f-  $999.999  +  $f 

PROBLEMS. 

1.  A  wagon  cost  $25.37,  a  horse  $180.90,  a  set  of  harness 
$10.05,  a  mowing  machine  $85.37,  a  cow  $28.08,  and  a  pair 
of  mules  $225.75.     What  was  the  entire  cost? 

2.  If  I  purchase  a  penknife  for  87-J-  cents,  a  quire  of  paper 
for  25  cents,  a  bottle  of  ink  for  18^  cents,  and  a  box  of  pens 
for  62^  cents,  how  much  money  will  pay  the  bill? 

3.  Add  ten  thousand   and   one  millionth,   four  hundred- 
thousandths,  ninety-six  hundredths,  forty-seven  million  sixty 
thousand  and  eight  billionths. 


136  PRACTICAL   ARITHMETIC 

4.  Five  pieces  of  silver  weigh  as  follows  :  .33  pounds,  1.275 
pounds,  .5  pounds,  2.375,  and  1.324  pounds.     How  much  do 
the  five  pieces  weigh  ? 

5.  Find  the  sum  of  2  decimal  units  of  the  2d  order,  2^  of 
the  3d  order,  4^  of  the  4th  order,  3£  of  the  5th  order,  5-J-J-  of 
the  6th  order,  and  94  of  the  7th  order. 

J  o 


SUBTRACTION. 

1.  What  is  the  difference  between  •£$  and  ^  ?     Between  .9 
and  .7  ? 

2.  What  is  the  value  of  .94  —  .02  ?     Of  .94  —  .06  ? 

Obviously,  the  subtraction  of  decimals  and  the  subtrac- 
tion of  integers  are  governed  by  the  same  principles. 

EXERCISES. 

1.  From  86.71  take  27.004. 

Process.  Explanation. 

86  710  1^e  numbers  being  written  with  the  decimal  points  in  a 

97  004  vertical  column,  the  digits  of  like  orders  fall  in  the  same 

column,  and  the  subtraction  proceeds  as  in  the  case  of  in- 
59.706  tegers.  The  7  tenths  must  be  separated  from  the  9  units  by 

the  decimal  point. 

RULE. 

"Write  the  decimals  so  that  the  decimal  points  will  fall  in 
a  vertical  column ;  subtract  as  in  subtraction  of  integers, 
and  place  a  point  in  the  difference  under  the  column  of 
points. 

2.  Subtract: 

(1.)  (2.)  (3.) 

82.19        274.684        8468.3684 
14.21        217.423        1764.8342 


SUBTRACTION  OF  DECIMAL  FRACTIONS  137 

(4.)  (5.)  (6.) 

8.4062  4256.85  $18.86 

.6434  .00564  13.685 

3.  Subtract  284.7654  from  321.07659. 

4.  Subtract  17.2398  from  27.06. 

5.  Subtract  29.9189  from  240.775. 

6.  Subtract  70.2574  from  365.71. 

7.  Subtract  .006  from  .00609. 

8.  Subtract  204.01  from  889.009. 

9.  Subtract  89.009  from  204.01. 

10.  Subtract  8.0999  from  1000. 

11.  Subtract  24.869  from  36. 

12.  Subtract  84  teu-millionths  from  84  millionths. 

13.  Subtract  9  dollars  37  cents  5  mills  from  27  dollars 
cents. 

14.  Subtract  406.375  from  2370.001. 

15.  Subtract  .074532  from  1.0003246. 

16.  Subtract  ffi  from  10.333J, 

17.  Find  the  value  of: 

1.  $125.75  — $41.095.    4.  $107.003  —  $.479. 

2.  $1.375  — $.88.  5.  5  —  5  hundred-millionths. 

3.  $2349  —  $29.33.       6.  3.625  —  1.5625. 


PROBLEMS. 

1.  A  man's  income  is  $3000  a  year;  he  spends  $487.50. 
How  much  does  he  lay  up  ? 

2.  I  bought  a  farm  for  $2560 ;  paid  at  one  time  $1046, 
and  at  another  time  $807.87.     What  remains  unpaid? 

3.  A  gentleman's  income  was  $1 2,384.1 6,  and  his  expenses 
the  same  year  were  $9864.18.     How  much  of  his  income  was 
left? 


138  PRACTICAL  ARITHMETIC 

4.  A  man  did  .37  of  a  piece  of  work  the  first  day  and  .33 
of  it  the  second  day.     What  part  of  the  work  was  left  for 
him  to  do  the  third  day  ? 

5.  John  walks  3.475  miles  and  James  3.8005  miles.    Which 
walks  the  farther,  and  how  much  ? 

6.  Simplify  : 

1.  8.763  —  4.12  +  78.326  —  68.0816. 

2.  ($.871  _|_  $143)  _  ($5>10  +  ^75^ 

3.  (155.006  —  .32)  -f  (80.0032  +  55.1). 

4.  12.07  —  11.432765. 

$-97  -  ($£  +  $.621  +  $|). 


MULTIPLICATION. 

!•  A  X  rk  =  what?     -3  X  .07  ==  what? 

2.  How  many  decimal  places  in  both  factors  ? 

3.  How  many  decimal  places  in  their  product  ? 

PRINCIPLE. 

The  product  of  decimal  factors  has  as  many  decimal 
places  as  the  factors. 

EXERCISES. 

1.  Multiply  9.06  by  .045. 

Explanation. 

9.06  =  fft  ;  .045  =  Tfe  ;  foe  x  ^  =  _y^  Or, 
since  the  factors  have  2  -f-  3  or  five  decimal  places,  the 
product  must  have  five  decimal  places  (Principle). 

.40770 

RULE. 

Multiply  -without  regarding  the  decimal  point,  but  in  the 
product  point  off  from  the  right  as  many  places  for  deci- 
mals as  there  are  decimal  places  in  the  factors. 

NOTE.  —  Should  the  result  of  a  multiplication  not  contain  as  many 
figures  as  the  factors  contain  decimal  places,  we  must  supply  the  deficiency 
by  prefixing  ciphers,  as  in  .02  X  -003  =  .00006. 


MULTIPLICATION  OF  DECIMAL  FRACTIONS        139 


2.  What  is  the  value  of : 

1.  13.2  X  2.475. 

2.  .132  X  2.475. 

3.  .236  X  12.13. 

4.  9.06  X  .045.. 

5.  .008  X  751.1. 

6.  70  X  387.45. 

7.  70.07  X  387.45. 

8.  4.2  X  .065. 

9.  2000  X  .075. 

10.  .436  X  .46. 

11.  .579  X  .035. 

12.  3.94  X  3.84. 

13.  5384  X  .0064. 

14.  .014  X  6.2  X  .007. 

15.  200  X  3|  X  .006. 

16.  947.36  X  .00423. 

17.  6|  X  7f  X  .81 

18.  .305  X  .00046. 

19.  10000  X  8.6213. 


20.  8.47  X  9.432. 

21.  .84  X  9.60. 

22.  3.468  X  2.008. 

23.  81  x  5.076. 

24.  28.8  X  41 

25.  8.375  X  61 

26.  |  X  2.5. 

27.  1561  X  .625. 

28.  1.776  X  .24. 

29.  1.603  X  2.564. 

30.  .0069  X  95.6. 

31.  2000  X  .075. 

32.  8000  X  .0755. 

33.  .785  X  .0191. 

34.  .00432  x  .00037. 

35.  81  X  .071  x  10. 

36.  .37  X  10000. 

37.  161  X  14.55. 

38.  277|  X  12.004. 


39.  3  hundredths  X  3  thousandths. 
40.  Four  hundred  thousand  two  hundred  sixty-eight 
ten-millionths  by  two  hundred  sixty  and  two 
hundred  seventy-five  thousandths. 


THE   DECIMAL   POINT   AS   A   MULTIPLIER. 

1.  .0004  X  10  =  what?    0.004  X  10  =  what?    00.04  X 
10  ==  what?     000.4  X  10  =  what? 

2.  Since  .0004  X  10  =  0.004  and  0.004  X  10  =  00.04, 
how  does  the  decimal  point  become  a  multiplier? 

3.  To  become  a  multiplier,  does  it  move  toward  the  right 
or  the  left? 


140  PRACTICAL  ARITHMETIC 

4.  Its  removal  one  place  to  the  right  multiplies  the  number 
by  what  ?  Its  removal  two  places  multiplies  by  what  ?  Three 

places  ? 

PRINCIPLE. 

Every  removal  of  the  point  one  place  toward  the  right 
multiplies  the  number  by  ten. 

RULE. 

To  multiply  by  a  number  consisting  of  1  with  ciphers  an- 
nexed, remove  the  decimal  point  as  many  places  towards 
the  right  as  there  are  ciphers  in  the  multiplier. 

EXERCISES. 

1.  Multiply  .394  by  100. 

Process.  Explanation. 

~»    .  Since  the  multiplier  is  one  with  two  ciphers  annexed,  we 

remove  the  decimal  point  two  places  towards  the  right,  and 
have  39  and  4  tenths  as  product. 

2.  Multiply: 

1.  8.7  by  10.  7.  9.2  by  10. 

2.  .0069  by  10.  8.  7.49  by  100. 

3.  95.6  by  100.  9.  .036  by  100. 

4.  .0453  by  100.  10.  854.3  by  1000. 

5.  4.069  by  1000.  11.  1.00182  by  10,000. 

6.  .000094  by  10,000.  12.  76.541  by  1,000,000. 

PROBLEMS. 
1.  Find  the  value  of: 

1.  57  horses,  at  $86.375  each. 

2.  200  barrels  of  flour,  at  $8.53^-  each. 

3.  251  yards  of  cloth,  at  $5-1-  a  yard. 

4.  236  bushels  of  oats,  at  $.515  a  bushel. 

5.  36f  bushels  of  clover  seed,  at  $4.52  a  bushel. 

6.  1000  pounds  of  wool,  at  $.375  per  pound. 

7.  280  barrels  of  apples,  at  $3f  a  barrel. 


DIVISION  OF  DECIMAL  FRACTIONS  141 

8.  100  cords  of  wood,  at  $5.47  a  cord. 

9.  305^  acres  of  land,  at  $82f  an  acre. 

2.  A   lady  made  the  following   purchases  :    47  yards  of 
sheeting,  at  $.14^  per  yard ;  9  yards  of  ribbon,  at  $.45-|-  per 
yard ;  38  yards  of  silk,  at  $3.46  per  yard.     What  did  her 
purchases  cost  her  ? 

3.  Multiply  six  hundred  twenty-five  ten-millionths  by  three 
and  eight  thousandths. 

DIVISION. 

1.  .6  X  -9  =  what? 

Since  one  factor  of  .54  is  .6,  what  is  the  other  factor  ? 

Since  .54  -r-  .6  =  .9,  how  does  the  number  of  decimal 
places  in  the  dividend  compare  with  the  number  in  the  divisor 
and  quotient  ? 

2.  .054  =  .09  X  .6. 

Assuming  .054  to  be  a  dividend  and  .09  to  be  a  divisor, 
what  is  the  quotient? 

Since  the  dividend  has  3  decimal  places  and  the  divisor  2, 
how  can  you  operate  with  3  and  2  to  find  the  number  of 
places  in  the  quotient? 

PRINCIPLE. 

The  number  of  the  decimal  places  in  the  quotient  equals 
the  number  of  places  in  the  dividend  minus  the  number  in 

the  divisor. 

EXERCISES. 

1.  Divide  82.32  by  2.1. 

Process.  Explanation. 

2.1  )  82.32  (  39.2  2.1  =  ft,  divisor ;  82.32  =  -^3_2  .  _8_?jL2  x  |o 

(J3  =  -3T9o2-  =  39.2.     Or,  dividing  without  regard  to 

-j  QO  the  decimal    point,   we   have    392  as  quotient. 

-.  £Q  Since  the  dividend  has  two  decimal  places  and 
the  divisor  one,  the  quotient  has  one;  hence  the 

42  quotient  sought  is  39.2. 

42 


142  PRACTICAL  ARITHMETIC 

BULB. 

Divide  without  regard  to  the  decimal  point,  but  finally 
point  off  from  the  right  of  the  quotient  as  many  figures  for 
decimals  as  the  number  of  decimal  places  in  the  dividend 
exceeds  the  number  of  those  in  the  divisor. 

NOTES. — 1.  "When  the  'quotient  does  not  contain  as  many  figures  for 
decimals  as  the  rule  requires,  supply  the  deficiency  by  prefixing  ciphers. 

2.  Before  beginning  to  divide,  it  is  best  to  make  the  number  of  decimal 
places  in  the  dividend  at  least  equal  to  the  number  of  decimal  places  in  the 
divisor. 

3.  When  the  process  of  division  has  used  only  as  many  decimal  places 
of  the  dividend  as  equal  the  number  of  decimal  places  of  the  divisor,  the 
quotient  will  be  an  integer. 

2.  Divide: 

1.  21.6  by  .006  (Apply  Notes  2  and  3). 

2.  .4913  by  1.7.  14.  3.2572  by  3.4. 

3.  2.1952  by  .028.  15.  467.37  by  100. 

4.  .5964  by  35  (Note  1).  16.  .003125  by  .125. 

5.  26.01  by  51.  17.  .03759  by  .01253. 

6.  .456  by  .06.  18.  .13  by  .026  (2  and  3). 

7.  4375  by  .25  (Note  2).  19.  .75  by  .025. 

8.  89.756  by  8.  20.  7  by  .007. 

9.  36.792  by  4.2.  21.  .4  by  .008. 

10.  44.98  by  1.3.  22.  .005  by  .0015. 

11.  .0002  by  .02.  23.  .0003  by  3.75. 

12.  325.72  by  34.  24.  .018  by  3600. 

13.  10,864.2  by  5432.1.  25.  2.0064  by  2.09. 

3.  Divide : 

1.  1235.434256  by  20.074. 

2.  195.388698  by  6.0708. 

3.  273.2879688  by  6.0708. 

4.  3.859243392  by  3.5702. 

5.  .00020596611  by  .03507. 

6.  625  ten-thousandths  by  25  millionths. 


DIVISION  OF  DECIMAL  FRACTIONS  143 

THE    DECIMAL   POINT   AS   A   DIVISOR. 

1.  4000  -s-  10  =  what?     400.0  -7-  10  =  what?     40.00  ~- 
10  =  what?     4.000  -5-  10  =  what? 

2.  Since  4000  -r- 10  =  400.0  and  400.0  -s-  10  =  40.00,  how 
does  the  decimal  point  become  a  divisor  ? 

3.  To   become   a  divisor,  does   the  decimal   point    move 
towards  the  right  or  the  left  ? 

4.  Its  removal  one  place  to  the  left  divides  the  number  by 
what?     Its  removal  two  places  divides  the  number  by  what? 
Three  places? 

PRINCIPLE. 

Every  removal  of  the  point  one  place  toward  the  left 
divides  the  number  by  ten. 

RULE. 

To  divide  by  a  number  consisting  of  1  with  ciphers  an- 
nexed, remove  the  decimal  point  as  many  places  toward 
the  left  as  there  are  ciphers  in  the  divisor. 

EXERCISES. 

1.  Divide  48.26  by  100. 

Process.  Explanation. 

4826  Since  the  divisor  is  1  with  two  ciphers  annexed,  we  re- 

move the  decimal  point  two  places  toward  the  left,  and  have 
.4826  as  quotient. 

2.  Divide: 

1.  534.79  by  100.  6.  4956.74  by  10,000. 

2.  492.568  by  1000.  7.  .038649  by  100,000. 

3.  24.9653  by  1000.  8.  82.253  by  1,000,000. 

4.  5.908  by  100.  9.  $9.391  by  10. 

5.  ,07156  by  1000.  10.  785.437  by  10,000. 


144  PRACTICAL   ARITHMETIC 

PROBLEMS. 

1.  Find  the  value  of  a  single  one  if: 

1.  144  eggs  cost  $2.88. 

2.  20  francs  =  $3.86. 

3.  20  shillings  =  $4.8665. 

4.  25  dress  patterns  —  102.50  yards. 

5.  125  bushels  of  oats  cost  $36.50. 

6.  TOO  acres  of  land  cost  $3156J. 

7.  72.50  C.  cigars  cost  $84.10. 

8.  1.440  M.  bricks  cost  $10.44. 

9.  22  days'  work  =  $29.70. 

10.  .62  of  a  ton  of  hay  cost  $11.47. 

11.  7|  acres  of  land  cost  $70.125. 

12.  .7}  yards  of  cloth  cost  $.73625. 

13.  5  weeks'  provisions  cost  $47.31  J. 

2.  Find  the  cost  of  8.25  tons  of  hay  when  2.2  tons  cost 
$311 

3.  Find  the  value  of  (6.25  -f-  3J-)  -r-  (3J-  —  .275). 


SHORT    PROCESSES. 
When  the  Multiplier  approximates  1OO,  1OOO,  etc. 

1.  How  much  less  than  10  times  a  number  is  9  times  that 
number  ? 

2.  How  much  less  than  100  times  a  number  is  98  times 
that  number  ? 

3.  Multiply  4965  by  99. 

Process.  Explanation. 

496,500  496,500  =  100  times  4965. 

4965  4965  —      1  time   4965. 


491,535  491,535  =    99  times  4965. 


SHORT  PROCESSES  145 

4.  Multiply : 

1.  4993  by  99.      6.  597,076  by  995. 

2.  4967  by  98.      7.  575,854  by  98. 

3.  59,678  by  999.    8.  954,367  by  96. 

4.  98,849  by  98.     9.  697,547  by  996. 

5.  457,836  by  997.   10.  5,064,367  by  97. 

5.  What  cost  496  bushels  of  wheat  at  $.98  per  bushel? 

6.  Find  the  cost  of  240  acres  of  land  at  $96  per  acre? 

7.  What  is  the  value  of  .0755  X  997? 

8.  2484  pounds  of  tea  at  $.96  =  what? 

9.  Find  the  value  of 

When  One  Part  of  the  Multiplier  is  an  Exact  Divisor  of 
Another  Part. 

1 .  6  is  how  many  times  3  ? 

2.  When  you  have  taken  a  number  3  times,  how  many 
times  that  product  will  6  times  the  number  be? 

3.  Multiply  1728  by  63. 

Process.  Explanation. 

1728  6  tens  =  2  tens  X  3.     1728  X  3  =  5184  units.     6184  X 

g3  2  tens  =  10,368  tens.     Adding  the  two  products  we  have 

"5184  108,864. 

10368 


NOTE. — Be  careful  in  placing  the  first  figure  of  each 


108,864  partial  product. 

4.  Multiply  : 

1.  4795  by  124  [12  =  4  X  3].   9.  7495  by  735. 

2.  4936  by  93  [9  =  3  X  3].  10.  5349  by  927. 

3.  7935  by  123.  11.  23,894  by  756. 

4.  25,384  by  142.  12.  47,523  by  918. 

5.  39,764  by  246.  13.  47,596  by  2505. 

6.  79,546  by  328.  14.  39,864  by  8024. 

7.  57,324  by  217.  15.  49,975  by  9045. 

8.  4793  by  945  [45  =  5  X  9].  16.  35,656  by  642. 

10 


146 


PRACTICAL    ARITHMETIC 


To  Multiply 


Process. 

4)189800 
47450 


Process. 
1900 

4 

76.00 

Process. 

3)189900 
6330Q 

Process. 
1728 

3 

51.84 

Process. 

8 ) 569600 
71200 

Process. 
3.14156 

8_ 

.2513248 

Process. 

6)78.54 
J3.09 


or  Divide  when  the  Multiplier  or  Divisor  is  an 
Aliquot  Part  of  1O,  1OO,  or  1OOO. 

1.  Multiply  1898  by  25. 

Explanation. 

Since  25  =  i££,  we  multiply  by  100  and  divide  the 
product  by  4. 

2.  Divide  1900  by  25. 

Explanation. 

if  £  inverted  =r  T*T ;    hence  we  multiply  by  4  and 
point  off'  two  places  to  the  right 

3.  Multiply  1899  by  331 

Explanation. 

Since  33|  =  -MJ-&,  we  multiply  by  100  and  divide  the 
product  by  3. 

4.  Divide  1728  by  331 

Explanation. 

J-f^  inverted  =  T^ ;    hence  we  multiply  by  3  and 
point  off"  two  places  to  the  right. 

5.  Multiply  5696  by  121. 

Explanation. 

Since  12J  —  i-jp,  we  annex  two  ciphers  and  divide 
the  product  by  8. 

6.  Divide  3.14156  by  121 

Explanation. 

if  £  inverted  =  T§^ ;    hence  we  multiply  by  8  and 
point  off  five  plus  two  places  to  the  right. 

7.  Multiply  .7854  by  16|. 

Explanation. 

Since  16|  =  ^{p,  we  multiply  by  100  and  divide  the 
product  by  Q, 


SHORT  PROCESSES  147 

Process.  8.  Divide  1492  by  16f 

1 499 

Explanation. 
f» 

i$£  inverted  —  T§7 ;    hence  we  multiply  by  6  and 
89.52  point  oft'  two  places  to  the  right. 

25,  331,  12|,  and  16f  are  aliquot  parts  of  100,  i.e.,  they 
are  such  parts  as  exactly  divide  100. 

Other  aliquot  parts  of  100  may  be  dealt  with  similarly  ; 
also  aliquot  parts  of  10  and  1000,  etc. 


EXERCISES. 

1.  Multiply  : 

1.  2556  by  25.  6.  40002  by  16f. 

2.  7.36  by  50.  7.  205.59  by  166f. 

3.  72.06  by  33J.  8.  380.087  by  12J. 

4.  207.27  by  3£.  9.  5908  by  14f 

5.  9.087  by  333^.  10.  390.8  by  2f 

2.  Divide: 

1.  404  by  25.  6.  399099  by  333f 

2.  5005.09  by  2f  7.  7906.73  by  16|. 

3.  407.709  by  50.  8.  970008  by  166f. 

4.  33659  by  33f  9.  5227.38  by  12|. 

5.  9304.75  by  3£.  10.  470058  by  14f 


PROBLEMS. 
Finding  the  Cost  of  Articles  sold  by  the  1OO,  1OOO,  and  Ton. 

1.  If  100  articles  cost  a  certain  price,  how  many  times 
that  price  will  500  articles  cost? 

2.  If  1000  articles  cost  a  certain  price,  how  many  times 
that  price  will  6000  articles  cost? 

3.  How  many  articles  does  C,  represent  ?     M.7  how  many  ? 


148  PEACTICAL  ARITHMETIC 

Process.  4.  What  will  2673  feet  of  timber  cost  at 

26.73         $2.25  per  C.  ? 
2.25  Explanation. 

13365  2673  feet  ==  26.73  C.  feet.     Since  C.  feet  cost  $2.25, 

5346  26.73  C.  feet  cost  $2.25  X  26.73  =  $60.1425. 

5346 
$60.1425 

5.  What  is  the    value  of   1262   fence   pickets   at   $12^ 
per  M.  ? 

Suggestion  :  1262  pickets  =  1.262  M.  pickets. 

6.  I  sold  6000  cigars  at  $4.20  per  C.     Find  the  amount 
received  therefor. 

7.  I  paid  $10.44  for  1440.     What  was  the  price  per  M.  ? 
(1440  =  1.440  M.) 

8.  If  the  price  of  gas  be  $1.75  per  M.,  find  the  amount 
of  a  man's  bill  when  12,240  cubic  feet  have  been  consumed. 

9.  What  cost  23|  M.  feet  of  pine  at  $55? 

10.  What  cost  4|  M.  brick  at  $8  ? 

11.  A  contractor  furnished  the  following  materials  for  a 
house :  34,600  bricks  at  $9.60  per  M.,  7960  feet  of  lumber 
at  $16  per  M.,  9050  feet  of  flooring  at  $22.50  per  M.,  7600 
shingles  at  40  cents  per  C.    Find  the  total  cost  of  the  materials. 

12.  If  2000  pounds  cost  $10.00,  what  will  1000  pounds  cost? 

13.  At  $5.00  per  M.,  what  will  3200  pounds  cost? 

14.  2000  pounds  =  one  ton.     If  one  ton  of  iron  costs  $30, 
what  will  1000  pounds  cost?     What  will  4000  pounds  cost? 

15.  What  must  I  pay  for  7850  pounds  of  stone  at  $2.20 
a  ton? 

Process.  Explanation. 

2  )$2.20  20°0 lbs-  cost  $2-20- 

-,  -JQ  1000  lbs.  cost  fl.10. 

"  7850  lbs.  =  7.850  M. 

7-85Q  Since  M.  cost  $1.10,  7.850  M.  cost  $1.10  X  7.850  = 
$8.63500 


REVIEW  OF  DECIMAL  FRACTIONS  149 

1 6.  What  will  3426  pounds  of  plaster  cost  at  $3.48  per  ton  ? 

17.  How  much  must  be  paid  for  6745  pounds  of  clay  at 
$15.25  a  ton? 

18.  For  7890  pounds  of  hay  at  $16.60  per  ton? 

19.  For  27,936  pounds  of  fertilizer  at  $18.50  per  ton? 

20.  For  7330  pounds  of  wool  at  $5.50  per  ton  ? 

21.  For  9041  pounds  of  iron  at  $125  a  ton? 


REVIEW. 

Indicate  the  processes. 

1.  What  is  the  quotient  when  3  is  divided  by  3  thou- 
sandths ? 

2.  If  the  divisor  is  207,  dividend  4776,  quotient  23,  find 
the  remainder. 

3.  The  product  of  two  numbers  is  |-,  and  one  of  them  is 
^  of  2.     What  is  the  other  ? 

4.  Reduce  .094t  to  a  common  fraction. 

o 

5.  Reduce  |~f-|  to  a  decimal. 

6.  Add  3.5  tons,  2.25  tons,  5  tons,  5.486  tons,  2.986  tons, 
3.6  tons,  2.336  tons,  and  2.376  tons. 

7.  Add  .0273  and  -fff^. 

8.  Subtract  .00976  from  2.03. 

9.  Find  the  value  of  .21  of  f  X  50  X  .Ollf 

10.  When   pork  is  selling  at  $6.25  per  hundred- weight, 
how  much  can  be  bought  for  $725? 

11.  Divide  the  sum  of  six  thousandths  and  six  millionths 
by  their  difference.     Find  six  decimal  places. 

12.  Multiply  732.89  by  33J.     [Use  short  process.] 

13.  Multiply  92.5674  by  333£. 

14.  Divide  96.325  by  121 

15.  How  many  pounds  of  coffee  can  be  bought  for  $16.25 
if  5£  pounds  can  be  bought  for  $1.78J? 


150  PRACTICAL    ARITHMETIC 

16.  (2.04  -r-  17  +  .235  X  5000)  —  £  =  ? 

17.  Divide  250,000  by  .00005. 

18.  I  paid  .33  of  a  sum  of  money  for  a  slate,  .17   for 
a  book,  and  .375  for  a  pair  of  skates.     What  fractional  part 
of  the  sum  was  lelt? 

19.  Simplify  '       3?  —  j,  expressing  the  result  as  a 

decimal. 

20.  What  is  the  value  of  95,150  bricks  at  $7.25  per  M.? 

21.  Subtract  .0507009  from  .08. 

22.  Reduce  .15f  to  a  common  fraction. 

23.  Find  the  cost  of  560  pineapples  at  $13.35  per  C. 

24.  Multiply  39,864  by  3609.     [36  =  4  X  9.] 

25.  Add  -fa,  -|,  and  ^  ;  subtract  -fa  from  ^  of  |-|  ;  subtiv.ct 
the  second  result  from  the  first,  and  take  -J-  of  the  difference. 

26.  If  15  tons  of  hay  cost  $125.25,  what  will  35  tons 
cost? 

27.  A  long  ton  =  -?4M  of  a  short  ton.    Reduce  to  a  mixed 

O  _-  U  U  U 

decimal. 

28.  Write  as  decimals  and  as  simple  common   fractions  : 


29.  Find  the  result  of  1.76  X  49.647  -=-  .0088. 

SO.  Add  -|f,  -J|-,  -|f  .     Express  the  result  as  a  decimal. 

31.  What  cost  164,960  pounds  of  coal  at  $6.00  per  short 
ton? 

32.  Bought  100  sheep  at  $3.375  a  head,  and  sold  them  at 
$3.875.    What  did  I  gain  on  each,  and  on  the  whole  number  ? 

33.  What  would  7f  bales  of  cotton  cost,  each  bale  weighing 
537.5  pounds,  at  $.1  If  a  pound? 

34.  A  horse  and  bridle  are  worth  $178.50;  but  the  horse 
is  worth  20  times  the  bridle.     Find  the  value  of  each. 

35.  Reduce  .7708^  to  a  common  fraction. 

36.  Multiply  793.295  by  .0001. 


REVIEW  OF  DECIMAL  FRACTIONS  151 

37.  How  much  iron  in  89,276  pounds  of  ore  if  .72  of  it 
is  pure  iron  ? 

38.  An  agent  charged  $5.85  for  collecting  a  bill  of  $260. 
What  was  his  charge  per  dollar  ? 

39.  If  .35  of  a  share  in  a  mining  company  is  worth  $31.15, 
what  is  the  value  of  15  shares? 

40.  A  grocer  sold  8970  pounds  of  sugar  at  $4.75  a  hun- 
dred pounds.     How  much  did  he  receive  ? 

41.  A  farmer  exchanged  9  tons  of  hay  worth  $16.87^  a  ton 
for  oats  at  31^  cents  a  bushel.     How  many  bushels  did  he 
receive  ? 

42.  If  49  yards  of  broadcloth  cost  $251.1  2  J,  what  would 
be  the  price  per  yard  ? 

43.  If  one  acre  of  land  costs  $38.75,  how  much  can  be 
bought  for  $3560? 

44.  How  many  days'  work  at  $1.25  a  day  must  be  given 
for  6  cords  of  wood  worth  $4.12^-  a  cord? 

45.  Bought  a  roll  of  carpet  containing  82  yards  for  $45, 
and  sold  it  for  75  cents  a  yard.     Find  the  amount  of  profit? 

46.  Find  the  value  of  60&  +  49^  +•!&&•  +  6}  +  90|. 

47.  Find  the  cost  of  3700  cedar  rails  at  $5.75  per  C. 

48.  If  a  man  earns  $12^-  a  week  and  spends  $7-f  per  week, 
in  how  many  weeks  can  he  save  $500  ? 

49.  What  is  the  value  of  86,260  bricks  at  $7.50  per  M. 


50.  Express  as  a  decimal 


~  *)       C*  + 


^ 

51.  What   cost  49.76  pounds  of  raisins  at  12J   cents  a 
pound  ? 

52.  What  cost  65  yards  of  muslin  at  16|  cents  a  yard? 

53.  Find,  the  cost  of  85  bushels  of  apples  at  33^  cents 
a  bushel. 

54.  What  must  you  pay  for  50  pairs  of  gloves  at  125 
cents  a  pair? 


152  PRACTICAL   ARITHMETIC 


ACCOUNTS  AND   BILLS. 

DEFINITIONS. 

1.  A  Debt  is  that  which  one  person  owes  to  another,  whether 
money,  goods,  or  services. 

2.  A  Credit   is   that  which   is  due  from  one  person  to 
another ;  or,  that  which  is  paid  towards  cancelling  a  debt. 

3.  A  Debtor  is  the  person  who  owes. 

4.  A  Creditor  is  the  person  to  whom  a  debt  is  due. 

5.  An  Account  is  a  record  of  debts  and  credits. 

6.  The  Balance  of  an  account  is  the  difference  between 
the  sums  of  the  debts  and  credits. 

7.  A  Bill  describes  the  goods  sold  by  giving  quantity  and 
price. 

8.  The  Footing  of  a  bill  is  the  total  cost. 

9.  A  Receipt  acknowledges  the  payment  of  a  bill  at  its 
foot,  thus  :    "  Received  payment, 

"  JAMES  JOHNSON." 

Common  Abbreviations. 

@,       at.  Cwt.,  hundred  weight.         Mdse.,  merchandise. 


a/c  ,        account. 

Do., 

the  same. 

No.,      number. 

Acc't,  account. 

Doz., 

dozen. 

Pay't,  payment. 

Bal.,    balance. 

Dr., 

debtor. 

Pd.  ,      paid. 

Bbl.,    barrel. 

Fr't, 

freight. 

Per,      by 

Bo't,    bought. 

Hhd., 

hogshead. 

Rec'd,  received. 

Bu.,  bushel. 

Inst., 

this  month. 

Ult.,  last  month. 

Co.  ,     company. 

Int., 

interest. 

Yd.,      yard. 

Cr.,      creditor. 

Lb., 

p™ind. 

Yr.,      year. 

ACCOUNTS  AND  BILLS  153 

Bills  are  usually  written  in  the  following  form : 

LANCASTER,  PA.,  July  31,  1898. 
Bought  of  BAIR  &  STEINM AN. 


MR.  R.  B.  RISK, 


2500  ft.  Boards, 

@  $27.50  per  M. 

$68 

75 

1875  ft.     do. 

"     25.00    u    " 

46 

88 

1650  Laths, 

"         .32    "    C. 

5 

28 

1520  Pickets, 

"     15.00    "    M. 

22 

80 

7500  Shingles, 

"       6.50    "     " 

48 

75 

$192 

46 

Find  the  footings  of  the  following  bills : 

(i.) 

MR.  JOHN  TODD, 


CHICAGO,  ILL.,  Aug.  3,  1898. 
Bought  of  RIGGS  &  CARTER. 


25,000  ft.  Pine  Boards,  @  $15.00  perM. 

8,500  «  Plank, 
11,850  "  Scantling, 

4,970  "  Timber, 

6,398  "       do. 


"  9.50  "  " 

"  7.00  "  " 

"  3.25  "  " 

"  4.00  "  « 


* 


Received  payment, 

RIGGS  &  CARTER. 


NOTE. — Finding;  the  value  of  the  different  items  of  a  hill  is  called 
making  the  extensions." 


154  PRACTICAL    ARITHMETIC 

(2.) 
DR.  J.  C.  GOOD, 


TRENTON,  1ST.  J.,  Aug.  4,  1898. 

Bought  of  STEPHEN  SMITH. 


35  Ibs.  Coffee 

@$.30 

$ 

5  Ibs  Tea 

"     .50 

30  Ibs.  Mackerel 

"     .15 

5  gals.  Molasses 

"     .60 

20  Ibs.  Sugar 

"     .05J 

3  doz.  Eggs 

"     .18 

2  Ibs.  Cheese 

"     .10 

3  Ibs.  Butter 

«     .20 

$ 

(3.) 


MR.  SHERMAN  ROGERS, 


ATLANTA,  GA.,  Aug.  5,  1898. 
To  PAUL  R.  JONES,  Dr. 


To  48  bbl.  Pork                      @  $12.50 
"    138  bbl.  Flour                    "       7.15 
"    4  bbl.  Molasses,  169  gal.     "         .40 
"    30  firkins  Butter,  2200  Ib.   "         .17 
"    4  boxes  Raisins                   "       4.60 
"    4  bbl.  Kerosene,  164  gal.    "         .19 
"    30  doz.  cans  Fruit              "       2.50 
"    2  bundles  Tobacco              "         .40 
"    12  doz.  Spices                      "       1.12J 

i 

$ 

i 

DENOMINATE  NUMBERS  155 

Set  in  bill  form  the  following  purchases,  find  the  footings, 
and  assume  that  the  bills  were  paid  : 

4.  Mrs.  T.  N.  Butcher  bought  of  Hervey  Martin,  15  yd. 
of  carpet  @  $1.00;  50  yd.  of  muslin  @  12J  cts. ;   18  yd. 
of  calico  @  9J  cts. ;  5  pairs  of  hose  @  75  cts. ;  15  yd.  of 
gingham  @  11 J  cts. ;  and  25  yd.  of  Canton  flannel  @  10  J 
cts. 

5.  Mr.  D.  F.  Lovett  bought  of  S.  Q.  Lowrey,  8679  ft.  of 
hemlock  @  $13.85  per  M. ;  9640  ft.  of  flooring  @  $24.75  per 
M. ;  6709  ft.  of  pine  @  $50.00  per  M. ;  4926  ft.  of  oak  @ 
$35.00  per  M. ;  8457  ft.  of  ash  flooring  @  $40.00  per  M. 

6.  Mr.  H.  K.  Landman  bought  of  B.  A.  Gross,  47  bu.  of 
wheat  @  $.87  ;  60  bu.  of  corn  @  $.60;  50  bu.  of  oats  @ 
$.33;  30  cwt.  of  flour  @  $3.50;  160  bu.  of  bran  @  $.18; 
83  Ib.  of  corn  meal  @  $.05. 

7.  Mr.  John  Rodgers  bought  of  William  H.  Cartwright, 
100  Ib.  of  breakfast  bacon  @  $.10 ;  55  Ib.  of  lard  @  $.08 ; 
37J  Ib.  of  picnic  hams  @  $.06 ;  45  Ib.  of  tallow  @  $.05^ ; 
25  Ib.  of  creamery  butter  @  $.17;  10  doz.  Western  eggs  @ 
$.12J;    16  Ib.  of  fowls  (hens)  @  $.15;   5  Ib.  of  cheese  @ 
$.10J;  12J  ib.  of  Rio  coffee  @  $.171. 


DENOMINATE   NUMBERS. 

Denominate  Numbers  are  Simple  or  Compound. 

A  Compound  Denominate  Number  is  composed  of  units 
of  two  or  more  denominations  that  have  among  them  a  certain 
natural  relation ;  as,  4  feet  6  inches,  or  3  bushels  2  pecks  1 
quart. 

Compound  Denominate  Numbers  have  their  origin  in  the 
existence  of  the  various  Measures  in  common  use. 


156  PRACTICAL  ARITHMETIC 

The  Measure  of  Value  is  Money,  which  is  also  called 
Currency. 

1:  United  States  Money  consists  of  Coin  and  Paper 

Money.  Coin  is  called  Specie.  The  Coins  are : 

GOLD.  SILVER. 

The  Double  Eagle  ==  $20.00.  The  Dollar .       =  $1.00. 

Eagle                       =    10.00.  Half-Dollar       =      .50. 

Half-eagle               =      5.00.  Quarter-Dollar  =      .25. 

Quarter  eagle          =      2.50.  Dime                  =      .10. 

The  Nickel  Coin  =  $.05. 

The  Bronze  Coin  =    .01. 

Other  United  States  coins  found  in  circulation  are  not  now 
coined. 

Paper  money  is  issued  in  the  form  of  bills  whose  face  value 
is  one  dollar  and  upward. 

The  Unit  of  United  States  Money  is  the  Dollar. 
Table. 

10  Mills  (m.)  =  1  Cent  (ct.). 

10  Cents          =  1  Dime  (d.). 

10  Dimes         =  1  Dollar  ($). 

10  Dollars       =  1  Eagle  (E.). 

$       d.        cts.          m. 

1  =  10  =  100  =  1000. 

Scale:  10,  10,  10  (Decimal). 

2.  Canadian  Money  has  the  denominations  of  the  United 
States  money,  except  the  gold  coins,  which  are  the  Sovereign 
and  Half-Sovereign. 

3.  French  Money  has  the  following  denominations  :   Cen- 
time, Decime,  and  Franc. 

The  Unit  is  the  Franc. 

Table. 

10  Centimes  (ct.)  (son-teems)  =  1  Decime  (dc.). 
10  Decimes  (des-seems)  =  1  Franc  (fr.). 

Fr.      dc.        ct. 

1  =  10  =  100.  =  $0.193. 
Scale:  10,  10  (Decimal). 


REDUCTION  DESCENDING  157 

4.  English  or  Sterling-  Money  is  the  currency  of  Great 
Britain.     The  coins  are  : 

GOLD.  SILVER. 

The  Sovereign   =  20  shillings.  Crown  =  5  shillings. 

Half-Sovereign  =  10         u  Florin  =  2         " 

Guinea  =  21         "  Shilling. 

Six-penny  piece. 
Three-penny  piece. 
COPPER  :  Penny,  Half-penny,  Farthing  (four  things). 

The  Unit  is  the  Pound  or  Sovereign. 

Table. 

4  Farthings  (far.)  =  1  Penny  (d.). 
12  Pence  =  1  Shilling  (s.). 

20  Shillings  =  1  Pound  (£). 

£.       s.         d.        far. 

1  =  20  =  240  =  960  =  $4.8665. 

Ascending  Scale :  4,  12,  20. 


REDUCTION  DESCENDING. 

INDUCTIVE    STEPS. 

1.  Since  4  farthings  =  1  penny,  how  many  farthings  =  2 
pence  ?     3  pence  ?     4  pence  ?     5  pence  ? 

2.  Since  12  pence  =  1  shilling,  how  many  pence  =  6  shil- 
lings?    10  shillings?     12  shillings? 

3.  How  many  shillings  in  £2?     In  £2  6  shillings? 

4.  How  many  pence  in  £2  6s.  5d.  ? 

5.  £l  =  how  many  shillings? 

6.  Js.  =  how  many  pence? 

7.  £\  =  how  many  shillings  ? 

Reduction  descending  changes  a  denominate  number  from 
a  higher  to  a  lower  denomination  without  changing  its  value. 


158  PEACTICAL   ARITHMETIC 

EXERCISES. 

1.  Reduce  £4  12s.  8d.  to  farthings. 
Process.  Explanation. 

£4  12s.  8d.  Since  £1  =  20s.,  £4  =  80s.     80s.  +  12s.  =  92s. 

20  since   ls-  =  12d.,   92s.  =   92  x    12d.  =  1104d. 

1104d.  +  8d.  = 


-i  9  '  Since  Id.  =  4  far.,  1112d.  =  1112  X  4  far.  =  4448 

-I  -Jo* 

far. 

Hence  £4  12s.  8d.  =  4448  farthings. 

2.  Reduce  £f  to  pence. 
Process. 


l2d.  £ 

_  4  =  J^d.  =  133-|d.     Or,  £f  X  -^  X  -^  = 


4448  far. 

3.  Reduce  £^  to  integers  of  the  lower  denominations. 

Process. 

£f  =  f  of  20s.  =  ^s.  =  llfs. 
f  s.  =  I-  of  12d.  =  -3/d.  —  5id. 
•4d.  =  -fof4far.  =  4-  far. 
Hence,  £f  =  11s.  5d.  Of  far. 

RULE. 

Multiply  by  the  numbers  of  the  scale  in  reverse  order, 
beginning  with  that  number  that  makes  one  of  the  highest 
given  denomination;  and,  as  you  proceed,  add  to  the  pro- 
ducts the  given  numbers  of  lower  denominations. 

4.  Reduce  : 

1.  £24  6s.  to  shillings. 

2.  £40  9s.  6d.  to  farthings. 

3.  £35  6s.  8d.  to  pence. 

4.  7s.  6d.  2  far.  to  farthings. 

5.  £14  18s.  lid.  to  pence. 

6.  £92  15s,  8d.  2  far.  to  farthings,. 


REDUCTION  ASCENDING  159 

7.  £56  4s.  lOfd.  to  farthings. 

8.  £f  +  fs.  -f  }d.  to  pence. 

9.  £3.  5s.  7|d.  to  farthings. 

10.  fs.  to  farthing. 

11.  £f  to  pence. 

12.  £f-  to  farthings. 

13.  £f-  to  integers  of  lower  denominations. 

14.  £^  to  integers  of  lower  denominations. 

15.  £-f-  to  integers  of  lower  denominations. 

16.  $150  to  mills. 

17.  $17.28  to  cents. 

18.  10  eagles  to  mills. 

19.  19  francs  to  decimes. 

20.  19  francs  8  decimes  to  centimes. 

REDUCTION  ASCENDING. 

INDUCTIVE   STEPS. 

1.  How  many  pence  in  8  farthings?    In  12  farthings?     In 

24  far.  ? 

2.  How  many  shillings  in  12d.  ?     In  36d.  ?     In  108d.  ? 

3.  40  shillings  equal  how  many  pounds  ?     60  shillings  ? 

4.  If  you  take  £2  out  of  45s.,  how  many  shillings  remain  ? 

5.  If  you  reduce  45s.  to  pounds,  what  is  your  result? 

Reduction   ascending  changes  a  denominate  number  from 
a  lower  to  a  higher  denomination  without  changing  its  value. 

EXERCISES. 

1.  How  many  pounds  are  there  in  8365  pence? 
Process.  Explanation. 

12)8365d.  Since  12d.  =  Is.,  T^  of  the  number 

9Q  \  £07,3    _[_  1(1  pence  =  the  number  of  shillings.    Since 

^  '  20s.  =  £1,  -fa  of  the  number  of  shillings 

=  the  number  of  pounds.    Hence,  a365d. 
8365d,  ^  £34  17s.  Id,         =  £34  17s.  Id, 


160  PRACTICAL  ARITHMETIC 

2.  Reduce  |<L  to  the  fraction  of  a  pound. 
Process. 

f  d.  =  f  of  A-s.  = 

TOT8-  =  T0~8  °*  <£21 
Or, 

^d    V  —  V    *    —  *  * 
9Q*   X   12  A  ?0  —  ^432 
4 

RULE. 

Divide  by  the  numbers  of  the  scale,  beginning  with  the 
number  of  the  given  denomination  that  makes  one  of  the 
next  higher,  and  continuing  until  the  required  denomina- 
tion is  reached. 

3.  Reduce  |-s.  to  a  fraction  of  a  pound. 

4.  Reduce  ^  far.  to  a  fraction  of  a  shilling. 

5.  Change  495d.  to  units  of  higher  denominations. 

6.  Change  4257  shillings  to  pounds. 

7.  Reduce  4697d.  to  pounds. 

8.  Reduce  5967s.  to  pounds. 

9.  Reduce  6969  far.  to  pounds. 

10.  Reduce  4995d.  to  pounds. 

11.  Reduce  5796  far.  to  shillings. 

12.  Reduce  59,678  far.  to  pounds. 

13.  Reduce  £4  15s.  6d.  to  far. 

14.  Reduce  59,607  far.  to  pounds. 

15.  Reduce  25,392  far.  to  £,  s.,  d. 

16.  Reduce  £26  to  dollars. 

17.  Reduce  $567  to  pounds. 

18.  Reduce  $394.45  to  pounds. 

19.  Reduce  $48.60  to  pounds. 

20.  Reduce  £36  to  dollars. 

21.  Reduce  25,488  far.  to  £,  s.  and  d. 

22.  Reduce  $973.30  to  pounds. 


MEASURES  161 

23.  Reduce  $1,216,625  to  pounds. 

24.  Reduce  $1084.40  to  cents. 

25.  Reduce  3596  cents  to  dollars. 

26.  Reduce  48,567  mills  to  dollars. 

27.  Reduce  1930  francs  to  dollars. 

28.  Reduce  3846  francs  to  dollars. 

29.  Reduce  4856  dollars  to  francs. 

30.  Reduce  5968  centimes  to  francs. 

31.  Reduce  |-  centime  to  the  fraction  of  a  franc. 

32.  Reduce  £^  to  integers  of  lower  denominations. 


MEASURES. 

1.  Extension  has  three  dimensions, — length,  breadth,  and 
thickness. 

2.  These  dimensions  are  the  origin  of  three  different  kinds 
of  Measures, — Linear  Measures,  Surface  Measures,  and  Meas- 
ures of  Volume  or  Capacity. 


LINEAR   MEASURES. 

Linear  Measures  are  used  in  determining  lengths  and  dis- 
tances. 

Common  Linear  Measure. 

12  Inches  (in.)  =  1  Foot  (ft.). 
3  Feet  =  1  Yard  (yd.). 

5i  Yards ") 

4  Feet    }        «*  **(*•)• 
320  Rods  =  1  Mile  (mi.). 

40  Rods  =  1  Furlong  (fur.). 

8  Furlongs       =  1  Mile. 
mi.      rd.         yd.  ft.  in. 

1  =  320  =  1760  =  5280  =  63,360. 
Scale:  12,  3,  5|,  320. 

11 


162  PKACTICAL   ARITHMETIC 

Surveyors'  Linear  Measure. 

7.92  Inches    =  1  Link  (1.). 
25  Links      =  1  Rod  (rd.). 

4Kods   ) 

-mn  T  •   i     r  =  l  Cham  (ch.). 
100  Links  j 

80  Chains    =  1  Mile  (m.). 
mi.     ch.        rd.  1.  in. 

I  =  80  =  320  =  8000  =  63,360. 
Scale :  7.92,  25,  4,  80. 

The  following  have  special  uses  : 

3  Barleycorns  =  1  Inch.     Shoe-length  measure. 

4  Inches  —  1  Hand.     Horse-height  measure. 
6  Feet  =  1  Fathom.     Sea-depth  measure. 
3  Feet  =  1  Pace.  ) 

5  Paces  =  1  Rod.   }  ^ing  measure. 

1.15  Statute  Miles  =  1  Geographical,  or  Nautical  Mile. 

3  Geographical  Miles  =  1  League. 

60  Geographical  Miles  \  [of  Latitude,  or 

89.16  Statute  Miles  j  =          6gree  (  of  Longitude  at  the  Equatoi 

EXERCISES. 

1.  In  12  rd.  3  yd.  2  ft,  how  many  feet? 

2.  In  5  mi.  18  rd.  4  yd.  how  many  yards? 

3.  Change  7  rd.  5  ft.  6  in.  to  inches. 

4.  Reduce  5  miles  to  inches. 

5.  In  5  mi.  2  fur.  4  yd.  how  many  feet? 

6.  In  5J  yd.  of  ribbon  how  many  inches? 

7.  How  many  inches  in  one  mile  ? 

8.  Reduce  3  mi.  4  yd.  2  ft.  to  feet. 

9.  Reduce  13,769  ft.  to  mi.,  fur.,  rd.,  etc. 

10.  In  15,347  in.  how  many  yd.,  ft.,  etc.? 

11.  Change  250,497  ft.  to  miles,  etc. 

12.  Reduce  77,565  in.  to  miles,  etc. 

13.  Reduce  5  miles  to  links. 

14.  Reduce  9  miles  54  ch.  to  links, 


SURFACE  MEASURES 


163 


15.  Reduce  11  mi.  68  fathoms  to  fathoms. 

16.  Reduce  43,000  1.  to  miles. 

17.  Reduce  79,400  1.  to  miles. 

18.  Reduce  9968  fathoms  to  miles. 

19.  In  481,401,716  in.  how  many  degrees? 

20.  If  a  horse  is  15^-  hands  high,  find  his  height  in  feet. 

21.  Change  29,763  1.  to  higher  denominations. 

22.  How  many  inches  in  2  degrees  at  the  equator  ? 

23.  Reduce  677,653  in.  to  higher  denominations. 

24.  Reduce  7912  mi.  (the  diameter  of  the  earth)  to  inches 
and  to  paces. 

25.  A  ship  was  sailing  in    12^-  fathoms  of  water.     How 
deep  was  the  water  ? 


SURFACE   MEASURES. 

1.  An  angle  is  formed  by  two  straight  lines  drawn  from 
the  same  point. 


Angle. 


Right  Angles. 


2.  When  a  line  is  drawn  from  a  point  between  the  ends  of 
another  line,  making  the  two  angles  equal,  the  angles  are 
called  right  angles. 

3.  A  Rectangle  is  a  figure  having  four  sides  and  four 
right  angles. 


Rectangle. 


Square. 


4.  A  Square  is  a  rectangle  with  four  equal  sides. 


164                     P: 

A. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

PRACTICAL    ARITHMETIC 

5.  The  Area  of  a  surface  is  the 
number  of  square  units  it  contains. 


B 


The  area  of  the  rectangle  A.  B.  is  12 
squares;  they  may  be  square  inches,  square 
feet,  or  square  rods,  etc. 

Since  there  are  4  rows  of  3  squares  each, 
the  area,  12  squares,  is  denoted  by  the  product 
of  4  by  3 ;  i.  e. ,  by  the  product  of  the  length 
by  the  breadth.  • 

If  by  4,  four  squares  are  meant,  is  3  to  be 


considered  an  abstract  or  a  denominate  number  ? 


FORMULA   (Indicated  Process). 
Area  of  rectangle  =  Length  X  Breadth. 

NOTE. — The  rule  implies  that  the  length  and  breadth  are 
the  same  denomination. 


in 


Common  Square  Measure. 
Table. 

144    Square  Inches  (sq.  in.)  =      1  Square  Foot  (sq.  ft.). 


9    Square  Feet 
30J  Square  Yards 
160    Square  Kods 
640    Acres 


=      1  Square  Yard  (sq.  yd.). 

_   (  1  Square  Rod  (sq.  rd.). 

~  (  1  Perch. 
=      1  Acre  (A.). 

Jl  Square  Mile  (sq.  mi.). 
1  Section  of  Land. 


sq.  mi.    A  sq.  rd.  sq.ft.  sq.  in. 

1  =  640  =  102,400  =  27,878,400  ==  4,014,489,600. 

Scale:  144,  9,  30J,  160,  640. 
NOTE. — 40  Perches  =  1  Rood  (R.)  ;  4  Roods  =  1  Acre. 

Units. 

The  unit  for  land  is  the  acre  ;  for  plastering,  ceiling,  etc.,  it  is  the  sq.  yd.  ; 
for  paving,  glazing,  and  stone-cutting,  it  is  the  sq.  ft. ;  for  roofing,  flooring, 
and  slating,  it  is  a  square  10ft.  by  10ft.,  or  100  sq.ft. 


SURFACE  MEASURES  165 

Surveyors'    Square   Measure. 
Table. 

625  Square  Links  (sq.  1.)  =  1  Sq.  Rd.,  or  Perch  (P.). 

16  Square  Rods  —  1  Sq.  Chain. 

10  Square  Chains  =  1  Acre. 

640  Acres  =  1  Sq.  Mile. 

36  Sq.  Miles  =  1  Township  (Tp.). 

EXERCISES. 
Reduce : 

1.  54  A.  10  P.  to  perches  (sq.  rd.). 

2.  624  P.  to  sq.  yds. 

3.  5  A.  145  P.  to  sq.  ft. 

4.  250  P.  17  sq.  yd.  to  sq.  inches. 

5.  2  A.  70  P.  30  sq.  yd.  to  sq.  inches. 

6.  10  sq.  yd.  4  sq.  ft.  15  sq.  in.  to  sq.  inches. 

7.  3,737,796  sq.  ft.  to  acres,  etc. 

8.  3  A.  37  sq.  rd.  5  sq.  yd.  7  sq.  ft.  to  sq.  inches. 

9.  295,376  sq.  in.  to  units  of  higher  denominations. 

10.  31  sq.  mi.  48  A.  to  sq.  ch. 

11.  2,000,000  sq.  1.  to  acres. 

12.  10,800  sq.  ch.  to  sq.  miles. 

13.  -J  of  A.  to  sq.  inches. 

Process. 
30i  =  f  •      1  X  f  X  f  X  f  X  f  =  4,878,720  sq.  in. 

14.  %  of  A.  to  units  of  lower  denominations. 

Process. 

|  A.  —  |  of  160  sq.  rd.  =  124$  sq.  rd. 
£  sq.  rd.  =  f  of  ip-  sq.  yd.  =  13f  sq.  yd. 
f  sq.  yd.  =  f  of  9  sq.  ft.  =  4  sq.  ft. 
Hence  $  A.  =  124  sq.  rd.  13  sq.  yd.  4  sq.  ft. 


166  PRACTICAL  ARITHMETIC 

15.  £  A.  to  units  of  lower  denominations. 

16.  J-  sq.  rd.  to  lower  denominations. 

17.  -f  A.  to  sq.  chains. 

18.  J-  section  of  land  to  sq.  1. 

19.  25,373,896  sq.  1.  to  higher  denominations. 

20.  8  chains  to  sq.  rods. 

21.  1  sq.  mi.  4  sq.  ch.  53  sq.  1.  to  links. 

22.  47,916  sq.  ft.  to  higher  denominations. 

23.  871,200  sq.  ft.  to  acres. 

24.  89,794,172  sq.  in.  to  acres,  etc. 

25.  20,000  sq.  ft.  to  sq.  rods. 

PROBLEMS. 

Indicate  the  process  first,  and  abridge  the  work  by  cancellation. 

1.  A  rectangular  piece  of  land  is  40  rd.  long  and  12  rd. 
wide.     Find  the  acres  in  it. 

2.  A  floor  is  8  ft.  by  16  ft.     How  many  sq.  ft.  in  it? 

3.  A  ceiling  is  17  ft.  by  20  ft.     How  many  sq.  yds.  in  it? 

4.  The  side  of  a  square  is  6  feet.     Find  its  area. 

5.  A  table  is  6  ft.  by  2  ft.     Find  its  area  in  sq.  in. 

6.  A  garden  is  656  ft.  by  93  ft.     Find  the  sq.  rd.  in  it. 

7.  A  fence  surrounding  a  mile  race-course  is  6  ft.  high. 
How  many  sq.  yd.  in  it?     Find  the  cost  at  10  cents  a  sq.  yd. 

8.  A  room  is  20  by  30  feet.     How  much  will  it  cost  to 
carpet  the  room  with  carpet  1  yd.  wide  at  $1.00  per  yd.? 

9.  A  field  contains  12  acres  and  is  24  rds.  wide.     Find 
its  length. 

10.  I  bought  10  acres  of  land  at  $200  an  acre  and  sold  it 
at  8  cts.  a  sq.  ft.     Find  the  gain. 

1 1 .  Find  the  cost  of  a  piece  of  oil-cloth  25  feet  long  and 
16  feet  9  inches  wide,  at  95  cents  a  square  yard. 

12.  What  will  it  cost  to  carpet  a  room  18  feet  long  and  25  J 
feet  wide  at  $1.25  a  yard,  the  carpet  being  J  yd.  wide? 


SURFACE  MEASURES  167 

13.  A  school-room   measures  as  follows  :   Length,  72  ft. ; 
width,  221  ft.;  height,  16  ft.     Deduct  245  sq.  ft.  for  doors 
and  windows,  and  find  the  cost  of  plastering  at  16  J  cts.  per 
sq.  yd. 

14.  Show  the  difference  between  6  sq.  ft.  and  6  feet  square. 

15.  A  room  is  to  be  plastered  and  painted;  its  length  is 
20  ft.,  its  width  18  ft.,  its  height  12  feet;  the  rate  will  be 
33  J  cts.  per  sq.  yd.     Find  the  cost  of  the  work. 

16.  Find  the  value  of  a  field  180  rd.  long  and  94J  rd. 
wide  at  $18  an  acre. 

17.  How  many  yds.  of  carpeting,  3  ft.  6  in.  wide,  will  it 
take  to  cover  a  floor  21  ft.  wride  and  36  ft.  long,  the  carpet 
running  lengthwise? 

18.  A  room  measures  18  ft.  X  15  ft,  X  10  ft.     Find  the 
cost  of  papering  it  with  paper  24  in.  wide  at  $.85  a  roll,  8  yd. 
in  a  roll,  making  a  deduction  of  20  sq.  yd.  for  openings. 

19.  A  sidewalk  is  10  ft.  wide,  exclusive  of  the  curb,  and 
is  100  ft.  long.     How  many  4X8  bricks  in  it? 

20.  A  15  by  18  ft.  room  is  to  be  carpeted.     Which  will  be 
the   cheaper   way   to   run    yard-wide    strips,    lengthwise   or 
breadthwise? 

21.  A  room  is  18  ft,  wide  and  9  ft.  high.     After  deducting 
from  the  area  of  one  end  two  windows  6  ft.  X  4J  ft.,  find  the 
number  of  sq.  yd.  remaining  to  be  plastered. 

22.  A  rectangular  piece  of  land  1320  yds.  long  and  2  rods 
wide  was  taken   for  public  use.     How  much  was  due  the 
owner  at  $160  an  acre? 

23.  A  barn  is  roofed  with  shingles  put  6  in.  to  the  weather. 
Find  the  cost  at  $12  per  M.  if  the  roof  is  60  feet  long,  each 
side  being  32   feet,  the  first  course  along   the  eaves  being 
doubled.     [1000  shingles  to  110  sq.  feet.] 

24.  Find  the  cost  of  slating  a  roof  64  ft.  9  in.  long  and  45 
ft.  wide  at  $15.37^  per  square. 


168 


PRACTICAL  ARITHMETIC 


25.  How  many  bricks  will  pave  a  sidewalk  25  ft.  by  10  ft., 
a  brick  measuring  8  in.  X  4  in.  X  2  in.  ? 

26.  How  many  bricks,  set  on  end,  will  pave  a  sidewalk 
containing  one-half  the  last  area  ? 


MEASURES   OF   VOLUME. 

1.  A  Solid  has  length,  breadth,  and  thickness. 

2.  A  Rectangular  Solid  is  bounded  by  six  rectangular 
faces. 


3.  A  Cube  has  six  equal  faces  and  twelve  equal  edges. 

4.  Volume,  or  Solid  Contents,  of  a  body  is  the  number 
of  cubic  units  it  contains. 

5.  If  on  a  rectangle  of  12  sq.  ft.,  as  a  base,  we  erect  a  rec- 
tangular solid  5  feet  high,  the  structure  will  contain  3  times 

4,  or  12,  cubic  feet  for  each  of  the  5 
feet  of  height.  Hence,  the  volume  of 
the  solid  will  be  5  times  12,  or  60, 
cubic  feet. 


RULE. 

The  volume  of  a  rectangular  solid 
is  the  number  of  cubic  units  de- 
noted by  the  product  of  its  length, 
breadth,  and  thickness. 


\ 

\ 

v       N 

\ 

\ 

\ 

\ 

\: 

V 

\ 

\ 

\ 

\ 
\ 

\ 

\ 

\ 

\ 

\ 

NOTE.— The  rule  implies  that  the  three  dimensions  are  all  of  the  same 
denomination. 


MEASURES  OF  VOLUME  169 

Cubic  Measure. 
Table. 

1728  Cubic  Inches  (cu.  in.)  =  1  Cubic  Foot  (cu.  ft.). 
27  Cubic  Feet  =  1  Cubic  Yard  (cu.  yd.). 

16  Cubic  Feet  =  1  Cord  Foot  (cd.  ft.). 

8  Cord  Feet,  or  "j 

=  1  Cord  of  TVood  (cd.}. 
128  Cubic  Feet      j 

Scale:  1728,  16,  8,  and  1728,  27. 

EXERCISES. 
Reduce : 

1.  15  cu.  yd.  18  cu.  ft.  16  cu.  in.  to  cu.  inches. 

2.  730,960  cu.  in.  to  cu.  yd.,  etc. 

3.  32  cu.  ft.  114  cu.  in.  to  cu.  inches. 

4.  174,964  cd.  ft.  to  cords. 

5.  7680  cu.  ft.  to  cords. 

6.  2160  cu.  ft.  to  cu.  yd. 

7.  62,950  cu.  in.  to  cu.  ft. 

8.  3  cd.  ft.  8  cu.  ft.  to  cu.  ft. 

9.  78,976  cd.  ft.  to  cords. 

10.  8797  cu.  ft.  to  cords. 

11.  466  cd.  124  cu.  ft.  to  cu.  ft. 

12.  1216  cu.  ft.  to  cords. 

13.  19,528  cd.  ft.  to  cords. 

14.  988  cu.  ft.  to  cu.  yd. 

15.  %  cd.  to  units  of  lower  denominations. 

FORMULAS. 

1.  Volume  =  length  x  breadth  X  height  (thickness). 

Height         i 
2'  Thickness  J   ="  volume  •*•  length  x  breadth. 

3.  Breadth          =  volume  -=-  length  x  height. 

4.  Length  ==  volume  -*-  breadth  x  height. 


170 


PRACTICAL  ARITHMETIC 


UNITS. 

The  cubic  foot  for  bricklaying,  masonry,  and  hewn  timber. 
The  cubic  yard  for  embankments,  excavations,  and  masonry. 
A  cubic  yard  of  common  earth  is  sometimes  called  a  load. 
The  perch  of  stone,  16|-  ft.  long,  1-i-  ft.  wide,  and  1  ft.  high,  equal  to 
24f  cu.  ft.     It  is  customary,  however,  to  call  25  cu.  ft.  a  perch. 

Brick. — The  size  of  a  common  brick  is  8  X  4  X  2  in.,  and  for  ordinary 
calculation  it  is  sufficiently  accurate  to  reckon  27  bricks  to  the  cubic  foot, 
laid  dry,  or  22^  laid  in  mortar. 

Brickwork  is  generally  estimated  by  the  thousand  bricks.  In  esti- 
mating material,  allowance  is  made  for  openings  in  walls,  as  doors,  win- 
dows, etc. 

In  estimating  labor,  the  length  of  each  wall  is  measured  on  the  out- 
side, and  thus  each  corner  is  measured  twice 

Sometimes,  by  special  contract, 
an  allowance  is  made  for  one-half 
the  openings  and  corners. 

A  pile  of  wood  8  ft.  long,  4  ft. 
wide,  and  4  ft.  high  is  a  cord. 

A  cord  foot  is  one  foot  in  length 
of  such  a  pile ;  that  is,  1  ft.  long, 

4  ft.  wide,  and  4  ft.  high. 
1      2     34      5      6  '-  '•- 


PROBLEMS. 

1.  Find  the  volume  of: 

1.  A  block  of  marble  9  ft.  long,  5  ft.  wide,  3J  ft. 

thick. 

2.  A  cube  whose  edge  measures  2  ft.  9  in. 

3.  A  pile  of  wood  36  ft.  long,  6  ft,  high,  4  ft.  wide. 

Suggestion:   3fi  Y/8y  ''•     Cancel. 

4.  The  contents  of  a  box  measuring  18  in.  by  16  in. 

by  14  in. 

5.  A  cube  1  in.  long,  1  in.  wide,  1  in.  high. 

6.  The  earth  in  a  cellar  36  ft,  X  24  ft.  X  5J  ft. 

7.  Excavation  for  a  reservoir  80  ft,  by  60  ft.  by  10  ft. 


MEASURES  OF  VOLUME  J71 

8.  Excavation  for  a  cistern  measuring  15  ft.  by  14 

ft.  by  13  ft. 

9.  A  cube  whose  sides  are  each  6  in.  square. 
10.  A  brick  measuring  8  X  4  X  2  in. 

2.  Why  are  there  16  cu.  ft.  in  a  cord-foot  of  wood? 

3.  Find  the  perches  of  masonry  in  a  wall  38  ft.  by  4  ft. 
by  1J  ft. 

Suggestion:  3S  *.*±*  **.     Cancel. 

4.  Reduce  16  p.  14J  cu.  ft.  to  cu.  ft. 

5.  A  cube  whose  edge  is  9  yds.  equals  in  volume  how 
many  cubes  whose  edge  is  one  foot  ? 

6.  How  many  blocks  of  one  cubic  inch  can  be  sawed  from 
a  cube  of  7  ft.,  if  there  is  no  waste  in  sawing  ? 

7.  How  many  cords  in  a  pile  of  wood  measuring  72  ft. 
by  13ft.  by  9ft.? 

8.  Find  the  cost  of  digging  a  cellar  50  ft.  long,  40  ft. 
wide,  5J  ft.  deep,  at  $1J  per  cubic  yard? 

9.  How  many  cubic  feet  in  two-thirds  of  a  cubic  yard  ? 

10.  How  many  cd.  ft.  in  5  cords  of  wood? 

11.  How  many  cu.  yd.  of  air  in  a  room  24  ft.  by  18  ft.  by 
12ft.? 

12.  Find  the  number  of  loads  of  ashes  in  a  rectangular  pit 
measuring  12  ft.  by  4  ft.  by  2  ft.  6  in.  ? 

13.  Find  the  cost  of  building  a  stone  wall  55  rds.  long,  4J 
ft.  high,  and  1  yd.  thick,  at  $7.25  a  perch  ? 

1 4.  How  many  bricks  would  make  the  bulk  of  9  loads  ? 

15.  How  many  perches  of  masonry  in  a  wall  6  ft.  high,  2 
ft.  thick,  inclosing  a  plot  of  ground  40  rds.  square  ? 

16.  How  many  bricks,  laid  in  mortar,  will  build  a  house 
57  ft.  long,  45  ft.  wide,  and  30  ft.  high,  the  wall  to  be  1J  ft. 
thick;  deducting  for  15  windows,  8J  by  3|  ft.,  and  8  doors, 
7f  by  3J  ft.  wide? 


172  PRACTICAL  ARITHMETIC 

17.  What  will  it  cost  to  dig  a  cellar  35  ft.  long,  22  ft.  wide, 
and  6  ft.  deep,  at  $.25  for  the  removal  of  a  cubic  foot  of  earth  ? 

18.  A  pile  of  wood  containing  19^  cd.  is  66  ft.  long  and 
6  ft.  wide.     Find  its  height? 

19.  Prescribing  18  in.  for  the  height  and  6  ft.  for  the  length 
of  a  wagon  body,  how  wide  must  it  be  made  to  hold  a  load  ? 

20.  Find  the  length  of  a  wall,  1  ft.  thick  and  4  ft.  high, 
that  1000  bricks,  laid  in  mortar,  will  build. 

21.  A  river  30  ft.  deep  and  20  yd.  wide  flows  4  mi.  an 
hour.     How  many  cu.  ft.  of  water  pass  a  given  point  in  a 
minute  ? 

Board  Measure. 

1.  Boards  whose  thickness  is  one  inch  or  less  are  measured 
by  the  square  foot.     A  board  12  ft.  long,  1  ft.  wide,  1  in.  thick 
=  12  sq.  ft.  board  measure. 

2.  All  hewn  or  squared  lumber  is  estimated  and  sold  by 
board  measure,  thickness  beyond  one  inch  becoming  a  factor. 

EXERCISE. 

How  many  feet,  board  measure,  are  there  in  a  timber  40  ft. 
long,  9  in.  wide,  and  6  in.  thick? 

Process. 
20 

&  x       X     =  180  ft.  board  measure. 


Explanation. 

9  in.  =  T9s  ft.  Since  -4T°-  X  T92  —  the  number  of  board  feet  when  the 
thickness  is  one  inch,  6  times  that,  or  -4r°-  X  -fz  X  f  —  ^e  number  of  board 
feet  when  the  thickness  is  6  in.  By  cancelling  we  have  180  ft. 

FORMULA. 

Number  of  feet  board  measure  =  length  (feet)  X  "width 
(feet)  X  thickness  (inches). 

NOTE.  —  When  the  board  tapers  uniformly,  use  the  mean  width,  i.e., 
half  the  sum  of  the  two  end  widths. 


MEASURES   OF  VOLUME  173 

PROBLEMS. 

I.  Find  the  number  of  board  feet  in  the  following  pieces 
of  lumber : 

1.  16  ft.  by  9  in.  6.  15  ft.  by  18  in.  by  4  in. 

2.  15  ft.  by  10  in.  7.  35  ft.  by  15  in.  by  12  in. 

3.  12  ft.  by  11  in.  8.  12  ft.  by  5  in.  by  4  in. 

4.  11  ft.  by  12  in.  9.  40  ft.  by  9  in.  by  6  in. 

5.  14  ft.  by  16  in.  10.  16  ft.  by  8  in.  by  f  in. 

2.  Find  the  number  of  feet  in  a  joist  20  ft.  long,  8  in. 
wide,  and  4  in.  thick. 

3.  A  man  has  70  planks  measuring  16  ft.  by  8  in.  and  If 
in.  thick.     How  many  feet,  board  measure,  has  he  ? 

4.  Find  the  width  of  a  2-in.  plank  16  ft.  long  that  con- 
tains 50  ft.,  board  measure. 

5.  Find  the  contents  of  a  board  18  ft.  long,  1  ft.  8  in.  wide 
at  one  end  and  14  in.  at  the  other. 

6.  Find  the  cost  of  38  boards  16  ft.  long,  12J  in.  wide,  at 
$2f  per  C. 

7.  Find  the  cost  of  the  following,  at  $20  per  M. : 

160  boards  16  ft.  by  11  in. 
170  boards  15  ft,  by  10  in. 
70  plank  14  ft.  by  10  X  3  in. 
70  scantling  12  ft.  by  4  X  2  in. 
50  rafters  25  ft.  by  5  X  3  in. 

8.  How  many  feet,  board  measure,  are  there  in  a  plank  17 
ft.  long,  22  in.  wide  at  one  end,  13  in.  wide  at  the  other,  and 
3  in.  thick  ? 

9.  Find  the  cost  of  72  boards,  each  11  ft.  long,  16  in. 
wide,  and  f  in.  thick,  at  $16.50  per  M. 

10.  In  a  tapering  board,  11  ft.  long,  18  in.  wide  at  one  end, 
11  inches  wide  at  the  other,  and  J  in.  thick,  how  many  feet? 

II.  A  room  is  20  by  25  ft.    What  will  be  the  cost  of  floor- 


174  PRACTICAL  ARITHMETIC 

ing  with  1J  in.  flooring,  at  $24  per  M.,  allowing  one-eighth 
for  matching? 

12.  A  field  160  yds.  long  by  120  yds.  wide  is  to  be  enclosed 
with  a  fence  4  boards  high,  each  board  6  in.  wide.  Find  the 
cost  of  the  boards  at  $18  per  M. 

MEASURES    OP    CAPACITY. 

Liquid  Measure. 
Liquid  Measure  is  used  in  measuring  all  kinds  of  liquids. 

Table. 

4  Gills  (gi.)  =  1  Pint  (pt). 
2  Pints          =  1  Quart  (qt.). 
4  Quarts        =  1  Gallon  (gal.). 
gal.    qt.    pt.      gi. 
1  =  4  =  8  =  32. 
Scale :  4,  2,  4. 

1.  The  capacity  Of  cisterns,  reservoirs,  etc.,  is  commonly  expressed 
in  gallons  or  in  barrels. 

2.  The  standard  liquid  gallon  of  the  United  States  contains  231  cubic 
inches. 

3.  The  barrel  (bbl.),  as   a   measure  of  capacity,  is  estimated   at  31| 
gallons  ;    the  hogshead  (hhd.)   at   63   gallons ;    neither,  as   a  commercial 
vessel,  holds  a  fixed  quantity. 

4.  The  beer  gallon,  of  282  cubic  inches,  is  no  longer  in  use. 

EXERCISES. 
1.  Reduce  the  following  : 

1.  7  pt.  to  gills. 

2.  8  qt.  to  gills. 

3.  37  gal.  to  pints. 

4.  795  pt.  to  gallons. 

5.  4957  gi.  to  gallons. 

6.  4  gal.  5  qt,  1  pt.  4  gi.  to  gills. 

7.  5  bbl.  7  gal.  to  gills. 

8.  9560  gi.  to  barrels. 


MEASURES  OF  CAPACITY  175 

9.  f  gal.  to  the  fraction  of  a  barrel. 
10.  1  bbl.  l  gal.  1  pt.  to  pints. 

2.  How  many  cu.  in.  in  8  gal.  ? 

3.  How  many  gal.  in  4956  cu.  in.  ? 

PROBLEMS. 

1.  A  cistern  is  16  ft.  long,  12  ft.  wide,  and  9  ft.  deep.    How 
many  bbl.  of  water  does  it  hold? 

2.  How  many  hhd.  will  a  cistern  contain  that  is  12  ft.  long, 
7  ft.  wide,  and  2  ft.  10  in.  deep? 

3.  Find  the  cu.  in.  in  the  space  that  will  hold  63  bbl. 

4.  If  a  vat  contains  54,762  cu.  in.,  how  many  bbl.  of  water 
will  it  hold? 

5.  If  a  cu.  ft.  of  water  weighs  1000  ounces,  what  is  the 
weight  of  10  hhd.  of  water? 

6.  How  many  cu.  ft.  in  a  cistern  that  contains  100  hhd.  ? 

7.  A  cistern  8  ft.  long  by  6  ft.  wide  contains  5  ft.  of  water. 
How  many  gal.  of  water  are  therein? 

8.  How  deep  must  a  cistern  be  to  contain  40  hhd.  if  it  is 
14  ft.  long  by  6  ft,  wide? 

Apothecaries'  Liquid  Measure. 

Apothecaries'   Liquid  Measure    is    used    for  measuring 
liquids  required  by  medical  prescriptions. 

Table. 

t!0  Drops  (gtt )  or  minims  (tt\J  =  1  Fluid  drachm 
8  Fluid  drachms  ==  1  Fluid  ounce  ( 

16  Fluid  ounces  =  1  Pint  (O.). 

8  Pints  =  1  Gallon  (Cong.). 

Gtt.  is  from  the  Latin  gutta,  a  drop. 
Minim  is  from  the  Latin  minimus,  the  least. 
O.  is  from  the  Latin  octarius,  one-eighth. 
Cong,  is  from  the  Latin  congius,  gallon. 


176  PRACTICAL  ARITHMETIC 

EXERCISES. 
Reduce : 

1.  12  pints  to  fluid  drachms. 

2.  48  fluid  ounces  to  pints. 

3.  12  gal.  3  pt.  to  fluid  ounces. 

4.  51  pt.  to  gallons. 

5.  34  Cong.  3  O.  1  f  3  3  f  3  to  n\,. 

6.  1860  ni  to  gallons. 

7.  16  Cong.  6  O.  7  f  3  to  f  3. 

8.  27,408  f  3  to  Cong. 

9.  4  Cong.  2  O.  15  f  5  7  f  3  to  nt. 
10.  8,472,347  TT\,  to  Cong.,  etc. 

Dry  Measure. 

Dry  Measure  is  used  in  measuring  dry  substances,  such  as 
grain,  roots,  fruit,  etc. 

Table. 

2  Pints  (pt.)  =  1  quart  (qt.). 
8  Quarts         ==  1  Peck  (pk.). 
4  Pecks          =  1  Bushel  (bu.). 
bu.    pk.     qt.       pt. 
1  =  4  =  32  =  64. 

Scale :  2,  8,  4. 

A  standard  bushel  contains  2150.42  cubic  inches. 
7  cubic  feet  of  corn  in  the  ear  equals  3  bushels  of  shelled  corn. 

Liquid  and  Dry  Measures  Compared. 

1.  Since  1  liquid  gallon  =  231  cu.  in., 

1  liquid  quart  —  how  many  cu.  in.  ? 
1  liquid  pint  =  how  many  cu.  in.  ? 
1  liquid  gill  =  how  many  cu.  in.  ? 

2.  Since  256  dry  gills  =  1  bu.  =  2150.42  cu.  in., 

1  dry  gill  =  how  many  cu.  in.  ? 

1  dry  pint  =  how  many  cu.  in.  ? 

1  dry  quart  =  how  many  cu.  in.  ? 

1  dry  gallon  =  how  many  cu.  in.  ? 


MEASURES  OF  CAPACITY  177 


1  gal.   1  qt.    1  pi.    1  gi. 

3.  Liquid  Measure :  231,     57f ,     28|,     7^  cu.  in. 

4.  Dry  Measure :        268f ,  67£,     33f,     8-f    cu.  in. 


EXERCISES. 
Reduce : 

1.  12  pt.  to  quarts.  6.  25  bu.  3  pk.  7  qt.  to  quarts. 

2.  5  bu.  to  pecks.  7.  38  bu.  5  qt.  1  pt.  to  pints. 

3.  32  qt.  to  bushels.  8.  42  pt.  to  quarts. 

4.  16  qt.  1  pt.  to  pints.      9.  402  pt.  to  pecks. 

5.  899  qt.  to  bushels.  10.  19  pk.  7  qt.  1  pt.  to  pints. 


PROBLEMS. 

1.  How  many  cu.  in.  in  8  bu.  ?     21  bu.  ? 

2.  How  many  bu.  in  27,692  cu.  in.  ? 

3.  How  many  cu.  in.  in  a  bin  9  ft.  long,  8  ft.  wide,  and 
6  ft.  high? 

4.  How  many  bu.  will  a  bin  hold  that  is  10  ft.  long,  7  ft. 
wide,  and  7  ft.  high? 

5.  What  must  be  the  depth  of  a  bin  to  contain  250  bu. 
of  grain,  its  length  being  12  ft.  and  its  width  6  ft.  ? 

6.  What  must  be  the  length  of  a  bin  whose  width  is  6  ft. 
and  depth  4  J  ft.  to  contain  400  bu.  of  rye  ? 

7.  A  bin  8  ft.  long,  7  ft.  wide,  and  5  ft.  deep  is  }  full  of 
oats.     What  is  the  value  of  the  oats  at  $.25  a  bushel  ? 

8.  If  a  vessel  holds  700  gal.  of  water,  how  many  bu.  of 
grain  will  it  contain  ? 

9.  How  many  bu.  of  grain  will  a  bin  hold  that  is  9  ft. 
5  in.  long  by  3  ft.  6  in.  wide  by  7  ft.  deep  ? 

10.  If  rain  falls  to  the  depth  of  1J  in.,  how  many  cu.  in. 
fall  on  one  acre? 

11.  A  rectangular  box  exactly  contains  a  bu.     The  length 
of  the  box  is  16  in.,  its  width  15.5  in.     Find  its  depth. 

12 


178  PRACTICAL  ARITHMETIC 

12.  How  many  bu.  of  corn  will  a  bin  hold  that  is  8.5  ft. 
long,  4.25  ft.  wide,  and  7J  ft.  deep? 

13.  The  capacity  of  a  bin  is  1583.2  bu. :  its  length  is  9  ft. 
and  depth  9  ft.     Find  its  width. 

14.  How  many  cu.  ft.  in  600  bu.  of  wheat? 

15.  How   many  cu.   ft.   in   500   bu.  of  potatoes,  heaped 

measure  ? 

1  bu.  heaped  meas.  =  \  bu.  stricken  meas. 

16.  How  many  cu.  ft.  in  a  bushel? 

17.  How  many  gal.  in  a  cubic  foot? 

18.  A  bin  18  ft.  long,  4±  ft.  wide  contains  40  cu.  yd.    Find 
its  depth. 

19.  A  bin  measures  6  ft.  5  in.  by  3  ft.  9  in.  by  4  ft.  6  in. 
How    many   bushels  of  wheat   will   it   hold  ?     How   many 
bushels  of  corn  in  the  ear? 

MEASURES    OP   WEIGHT. 

Weight  is  the  measure  of  the  force  that  attracts  bodies  to 
the  earth. 

Avoirdupois  Weight. 

Avoirdupois  Weight  is  used  in  weighing  all  materials 
except  gold  and  silver. 

Table. 

16  Ounces  (oz.)        =  1  Pound  (lb.). 
100  Pounds  =  1  Hundred- weight  (cwt.). 

20  Hundred-weight  —  1  Ton  (T.). 

T.     cwt.       lb.  oz. 

1  =  20  =  2000  =  32,000. 

Scale :  16,  100,  20. 

The  pound  =  7000  grains. 
The  ounce  —  16  drachms. 
The  long  ton  =  2240  Ibs. 


MEASURES  OF  WEIGHT  179 

Measures  Much  Used. 

1  Firkin  (of  butter)  =    56  Ib. 

1  Cental  (of  grain,  flour)  =  100  Ib. 

1  Quintal  (of  dried  fish)  =  100  Ib. 

1  Keg  (of  nails)  =  100  Ib. 

1  Barrel  (of  flour)  =  196  Ib 

1  Barrel  (of  pork  or  beef)  =  200  Ib. 
1  Barrel  (of  salt  at  N.  Y.  Works)  =  280  Ib. 

1  Cask  (of  lime)  =  240  Ib. 

Pounds  in  a  Bushel. 


Wheat,  60  Ib. 

Beans,            60  Ib. 

Wheat  Bran, 

20  Ib. 

Eye,       56  Ib. 

Buckwheat,  42  Ib. 

Salt, 

56  Ib. 

Corn,      56  Ib. 

Flax  Seed,     56  Ib. 

Corn  Meal, 

50  Ib. 

Barley,  48  Ib. 

Hemp  Seed,  44  Ib. 

Corn  in  Ear, 

68  Ib. 

Oats,      32  Ib. 

Potatoes,       60  Ib. 

Clover  Seed, 

60  Ib. 

Peas,      60  Ib.          Onions,          60  Ib.         Timothy  Seed,  45  Ib. 
NOTE. — Slight  variations  from  the  above  exist  among  the  States. 

EXERCISES. 
Reduce : 

1.  3  T.  to  pounds.  4.  95,000  oz.  to  tons. 

2.  5  T.  4  cwt.  6  Ib.  to  pounds.        5.  2  T.  to  ounces. 

3.  10  T.  327  Ib.  to  pounds.  6.  10,406  Ib.  to  tons. 

PROBLEMS. 

1.  At  $.05  J  a  pound,  what  will  3  cwt.  of  sugar  cost? 

2.  What  will  4  J  Ib.  of  confections  cost,  at  $.04  J  per  ounce  ? 
,3.  At  $.08  per  Ib.,  what  are  8  cwt.  of  beef  worth  ? 

4.  How  many  grains  in  9  Ib.  ? 

5.  What  will  400  Ib.  of  coal  cost  at  $6.50  per  T.  ? 

6.  How  many  bushels  of  corn  meal  in  2  T.  ? 

7.  How  many  pounds  will  1000  bu.  of  oats  weigh? 

8.  How  many  T.  will  500  bu.  of  potatoes  weigh  ? 


180  PRACTICAL  ARITHMETIC 

9.  Find  how  much  more  160  bu.  of  wheat  weigh  than 
200  bu.  of  barley. 

10.  Find  the  cost  of  3024  Ib.  of  rye,  at  $.66  a  bu. 

11.  Find  the  value  of  5  firkins  of  butter,  at  $.17  per  Ib. 

12.  In  30,000  Ib.  of  pork  how  many  barrels  are  there? 

13.  What  is  the  weight  of  105  bu.  3J  pk.  of  potatoes? 

14.  If  a  dealer  has  27  long  tons  of  coal  and  sells  7780  Ib., 
how  many  short  tons  remain  ? 

15.  At  $10  a  barrel,  what  will  a  bag  of  flour  cost,  weighing 
59  Ib.  ? 

Troy  Weight. 

Troy  Weight  is  used  in  weighing  gold,  silver,  and  precious 

stones. 

Table. 

24  Grains  =  1  Pennyweight  (pwt.). 

20  Pennyweights  =  1  Ounce  (oz.). 
12  Ounces  =  1  Pound  (Ib.). 

Ib.      oz.      pwt.        gr. 

1  =  12  =  240  =  5760. 
Scale:  24,  20,  12. 

The  carat,  equal  to  4  grains,  is  commonly  used  in  weighing  precious 
stones.'  Carat,  as  the  unit  of  fineness  for  gold,  means  J?-  Gold  14  carats 
fine  is  ||  gold  and  \\  alloy. 

EXERCISES. 
1 .  Reduce : 

1.  9  pwt.  12  gr.  to  grains. 

2.  204  gr.  to  pennyweights. 

3.  70  Ib.  2  oz.  19  pwt.  16  gr.  to  grains. 

4.  4438  gr.  to  ounces. 

5.  5  Ib.  11  oz.  15  pwt.  10  gr.  to  grains. 

6.  8356  gr.  to  ounces. 

7.  150  pwt.  to  oz. 

8.  1  Ib.  to  grains. 


MEASURES  OF  WEIGHT  181 

2.  How  many  grains  in  a  pound  avoirdupois  ? 

3.  How  many  more  grains  in  a  pound  avoirdupois  than  in 
a  pound  troy  ? 

4.  Which  is  heavier,  and  how  much,  a  pound  of  lead  or  a 
pound  of  silver? 

5.  How  many  pounds  avoirdupois  are  there  in  185  troy 
pounds  ? 

6.  Find  the  value  in  troy  weight  of  9  Ib.  10  oz.  avoirdupois. 

7.  If  an  ounce  of  gold  is  worth  $25.75,  what  is  the  value 
of  a  pennyweight? 

8.  If  1898  sovereigns  weigh  40.6172  Ib.  troy,  how  many 
grains  does  one  sovereign  weigh  ? 

Apothecaries'  Weight. 

Apothecaries'  Weight  is  used  in  prescribing  and  mixing 
dry  medicines. 

Medicines  are  bought  and  sold  by  avoirdupois  weight. 

Table. 

20  Grains  (gr.)  =  1  Scruple  ®). 

3  Scruples        =  1  Drachm  (3). 

8  Drams  =  1  Ounce  (£). 

12  Ounces          =  1  Pound  (Ib.). 

Ib.       oz.      dr.       sc.         gr. 
1  =  12  =  96  =  288  =  5760. 
Scale :  20,  3,  8,  12. 

Scruple  :  Latin,  scrupulus,  a  little  stone. 
Drachm  :  Greek,  drachma,  a  piece  of  money. 
Ounce  :   Latin,  uncia,  one-twelfth. 

1.  1  oz.  Avoirdupois  equals  how  many  grains? 

2.  1  oz.  Troy  equals  how  many  grains  ? 

3.  1  oz.  Apothecaries'  equals  how  many  grains  ? 


182  PRACTICAL  ARITHMETIC 

EXEBCISES. 
Reduce : 

1.  969  to  drachms. 

2.  1445  to  pounds. 

3.  8  Ib.  65  to  drachms. 

4.  8163  to  pounds. 

5.  18  Ib.  63  43  19  to  scruples. 

6.  9523  to  pounds. 

7.  17  Ib.  53  29  17  gr.  to  grains. 

8.  105,840  gr.  to  pounds. 

PROBLEMS. 

1.  Express  4567  grains  apoth.  in  higher  units. 

2.  How  mauy  4-grain  pills  can  be  made  from  63  29  ? 

3.  How  many  ounces  of  quinine  will  be  required  to  make 
1440  2-grain  pills? 

4.  How  many  5-grain  pills  can  be  made  from  an  avoir- 
dupois pound  of  quinine  ? 

5.  Change  IS  53  19  16  gr.  to  the  fraction  of  a  pound. 

6.  Reduce  8  Ib.  avoirdupois  to  apothecaries'  weight. 

7.  How  much  does  a  druggist  gain  who  buys  15  Ib.  (avoir.) 
of  drugs  at  $2.75  per  pound  and  sells  the  same  at  $.20  per 
drachm,  apoth.  weight? 

8.  An  apothecary  bought  50  Ib.  8  oz.  of  opium  at  45  cents 
an  ounce,  and  sold  it  at  3  cents  a  scruple.     How  much  did  he 
gain? 

MEASURES   OF    TIME. 

Time  is  measured  by  centuries,  years,  months,  weeks,  days, 
hours,  minutes,  and  seconds. 

The  unit  is  the  day,  which  is  determined  by  the  revolution 
of  the  earth  on  its  axis.  The  year  is  determined  by  the  revo- 
lution of  the  earth  around  the  sun. 


MEASURES  OF  TIME  183 

Table. 

60  Seconds  (sec.)  =  1  Minute  (min. ). 
60  Minutes  =  1  Hour  (hr.). 

24  Hours  =  1  Day  (da.). 

7  Days  =  1  Week  (wk .). 

365  Days  =  1  Year  (yr.). 

366  Days  =  1  Leap  year. 
100  Years  =  1  Century  (cen.). 

yr.     mo.      da.          hr.  min.  sec. 

1  =  12  =  365  =  8760  =  525,600  =  31,536,000. 
Scale :  60,  60,  24,  365,  100. 

1.  A  year  is  called  Leap  Year  when  the  number  denoting 
it  is  divisible  by  4  and  not  by  100,  or  is  divisible  by  400. 

2.  A  year  is  called  a  Common  Year  when  the  number 
denoting  it  cannot  be  thus  divided. 

3.  The  Calendar  divides  the  year  into  weeks  and  months. 

Table. 

1.  January,         31       days,  Jan.  7.  July,  31  days,  July. 

2.  February,  28  or  29  days,  Feb.  8.  August,        31  days,  Aug. 

3.  March,  31       days,  Mar.  9.  September,  30  days,  Sept. 

4.  April,  30       days,  Apr.  10.  October,       31  days,  Oct. 

5.  May,  31       days,  May.  11.  November,  30  days,  Nov. 

6.  June,  30      days,  June.  12.  December,   31  days,  Dec. 

A  Useful  Stanza. 

Thirty  days  hath  September, 
April,  June,  and  November; 
All  the  rest  have  thirty-one, 
Excepting;  February  alone ; 
To  which  we  twenty-eight  assign, 
Till  leap  year  gives  it  twenty-nine. 

EXERCISES. 
Reduce : 

1.  One  day  to  seconds. 

2.  86,400  seconds  to  days. 


184  PRACTICAL  ARITHMETIC 

3.  One  week  to  minutes. 

4.  20,160  minutes  to  weeks. 

5.  4  da.  6  h.  56  min.  to  min. 

6.  5776  min.  to  days. 

7.  32  hr.  32  min.  41  sec.  to  seconds. 

8.  87,990  sec.  to  hours. 

PROBLEMS. 

1.  Express  49,796  sec.  in  higher  denominations. 

2.  Express  49,598  sec.  in  units  of  higher  orders. 

3.  How  many  days  from  April  10th  to  September  12th? 

4.  How  many  days  from  March  22d  to  July  17th? 

5.  Giving   three   months  to  each,  which  is  the  longer, 
summer  or  winter? 

6.  Which  of  these  are  leap  years:   1600,   1660,   1666, 
1700,  1776,  1790,  1794,  1800,  1898,  1900? 

7.  How  many  seconds  are  there  in  7  hr.  38  min.  49  sec.  ? 

8.  How  many  days  and  hours  in  -|  week? 

9.  Find  the  number  of  hours  in  a  week. 

10.  What  part  of  a  week  is  1  day  18  hours? 

11.  A  man  borrows  some  money  July  17  and  promises  to 
repay  it  in  60  days.     On  what  day  is  it  due  ? 

12.  How  many  leap  years  in  the  nineteenth  century? 

13.  How  many  hours  in  the  month  of  August? 

14.  If  you  read  French  30  min.  each  day  for  5  yr.,  how 
much  time  do  you  thus  spend  ? 

15.  On  what  day  will  J-  of  a  common  year  end?     f  of  a 
leap  year  ? 

CIRCULAR   MEASURE. 

Circular  Measure  is  used  to  measure  angles. 
A  Circle  is  a  figure  made  by  a  bounding  line  which  is 
everywhere  equally  distant  from  a  centre-point. 


CIRCULAR  MEASURE 


185 


The  Circumference  of  the  circle  is  the  bounding  line. 

An  Arc  is  any  part  of  the  circum- 
ference, as  BD. 

A  Quadrant  is  an  arc  equal  to  \  of 
the  circumference,  as  BE. 

A  Degree  is  -^-Q  of  the  circum- 
ference of  a  circle. 

An  angle  whose  sides  meet  at  the  centre  is 
measured  by  the  arc  included  between  its  sides. 
The  angle  BCD  is  measured  by  the  arc  BD.     A  right  angle  is  measured 
by  a  quadrant,  or  90°. 

Table. 

60  seconds  (")      =  1  Minute  ('). 
60  Minutes  =  1  Degree  (°). 

30  Degrees  =  1  Sign  (S.). 

12  Signs,  or  360°  —  1  Circumference  (C.). 
The  Sio-n  is  used  in  astronomical  calculations. 


5.  35°  48'  59"  to  seconds. 

6.  99,800"  to  degrees. 

7.  4  S.  29°  26'  33"  to  seconds. 
490,833"  to  signs. 


EXERCISES. 

1.  Reduce: 

1 .  56'  25"  to  seconds. 

2.  3830"  to  minutes. 

3.  23°  36'  to  minutes. 

4.  856'  to  degrees. 

2.  How  many  minutes  in  2  quadrants? 

3.  How  many  seconds  in  a  right  angle? 

4.  2  J  quadrants  are  what  part  of  a  circumference  ? 

5.  In  811,480"  how  many  signs? 

6.  In  what  time  does  a  fixed  point  in  the  earth's  surface 
pass  through  15°  15'  15"? 

7.  Where  is  a  degree  of  latitude  equal  to  60  geographical 
miles  ? 

8.  Sixty-nine  and  16  hundredths  statute  miles  equal  one 
degree  on  what  surface? 


186  PRACTICAL  ARITHMETIC 

MISCELLANEOUS    TABLES. 
Counting-. 

12  Units   =  1  Dozen.  12  Gross  =  1  Great  gross. 

12  Dozen  =  1  Gross.  20  Units  =  1  Score. 

Paper. 

24  Sheets  =  1  Quire.  2  Reams    =  1  Bundle. 

20  Quires  =  1  Ream.  5  Bundles  =  1  Bale. 

Books. 
A  book  composed  of  sheets  folded  in : 

2  leaves  is  a  folio.  12  leaves  is  a  duodecimo. 

4  leaves  is  a  quarto.  16  leaves  is  a  16mo. 

8  leaves  is  an  octavo.  18  leaves  is  an  18mo. 

EXERCISES. 
Reduce : 

1.  45  gross  of  crayons  to  units. 

2.  222  dozen  bottles  of  ink  to  gross. 

3.  12  great  gross  to  units. 

4.  3  reams  of  paper  to  quires. 

5.  5  bundles  of  paper  to  sheets. 

PROBLEMS. 

1 .  What  would  9600  sheets  of  foolscap  cost  at  $.25  per 
quire? 

2.  Find  the  cost  of  4  dozen  brushes  @  $.55  each. 

3.  A  dealer  bought  paper  at  $8  per  ream  and  sold  it  at 
$.30  a  quire.     Did  he  gain  or  lose,  and  how  much  ? 

4.  In  an  octavo  book  of  960  pages  how  many  sheets  ? 

5.  How  many  years  are  3  score  and  10? 

6.  How  many  sheets  in  2  bundles  1  ream  15  quires  10 
sheets? 

7.  How  many  units  in  8  gross  9  dozen  ? 


REDUCTION  OF  DENOMINATE  FRACTIONS          187 

8.  If  12  dozen  of  buttons  are  worth  $1.08,  what  are  11 
buttons  worth? 

9.  The  use  of  48  screws  per  day  implies  the  use  of  how 
many  gross  in  6  weeks? 

10.  500,000  copies  of  a  daily  newspaper  are  sold  on  an 
average  per  diem.  Reckoning  3  sheets  for  each  copy,  how 
many  reams  of  paper  are  used  in  a  month  ? 

REDUCTION  OF  DENOMINATE  FRACTIONS. 

SPECIAL   EXERCISES. 
Reduction  Descending-. 

1.  Reduce  f  of  a  rod  to  units  of  lower  denominations. 

Process.  Explanation. 

6.  of  if  yd.  =  ff  yd.  =  4f  yd.  6*  yd-  =  ¥  yd-    Since  l 

t  of  3  ft.  =  ¥ ft.  =  2>  ft.  •          J  =  55- *$£*£ I 

\  of  12  in.  =  V-  m.  =  1|  in.  3  f t  ?  f  yd  =  f  of  3  ft.  =  y. 

-f-  rd.  =  4  yd.  2  ft.  If  in.  ft.  =  2f  ft.    Since  1  ft.  =  12 

in.,  f  of  aft.  =}  of  12  in.  = 
if  in.  =  1^  in.  Hence  f  rd. 
=  4  yd.  2  ft.  If  in. 

2.  Reduce  .795  Ib.  Troy  to  units  of  lower  denominations. 

Process.  Explanation. 

.795  Ib.  Since  12  oz.  =1  Ib.,  12  times  the  number  of  pounds 

12  =  tbe  number  of  ounces;  12  times  .795  =  9.540  oz. 

9  540  Since  20  pwt.  =  1  oz.,  20  times  the  number  of  ounces  = 

20  the  number  of  pwt.     .540  X  12  =  10.8  pwt.     Hence 

190.800  .795  lb.=  9OZ.  10.8  pwt. 

3.  Reduce  -g-f-g-  gal-  to  gills. 

Process.  Explanation. 

—  V4v2v4  =  -*n  Since  l  gal'  =  4  qt- J  qt-  =  2  pt., 

?$$  '  9  8  and  1  pt.  =  4  gi.,  we  multiply  by  the 

numbers  of  the  scale,  4,  2,  4,  and  ob- 
tain by  cancellation  f  gills. 


188  PRACTICAL  ARITHMETIC 

EXERCISES. 

1.  Reduce : 

1.  -f£$  of.  a  bushel  to  the  fraction  of  a  pint. 

2.  -gJ-g-  da.  to  minutes. 

3.  f  rd.  to  yards,  feet,  and  inches. 

4.  .065  of  a  gallon  to  integers  of  lower  denominations. 

5.  ^  of  a  ton  to  lower  denominations. 

6.  y^j-  of  an  acre  to  lower  denominations. 

7.  .007  of  a  bushel  as  a  decimal  of  a  pint. 

8.  .796  of  a  Ib.  troy  to  lower  denominations. 

9.  £.686  to  lower  denominations. 

10.  .436  of  a  ream  to  integers. 

11.  .875  of  a  leap  year  to  integers. 

12.  .795  of  a  league  to  integers. 

13.  -29tf  cu-  yd.  to  lower  denominations. 

14.  .115625  Ib.  troy  to  integers. 

2.  Reduce: 

1.  yf^  bu.  to  pints.  9.   12^6o  T.  to  ounces. 

2.  £l71aa  to  pence.  10.   l8j[2Tr  mi.  to  inches. 

3.  -|  Ib.  troy  to  integers.  11.  -J  da.  to  integers. 

4.  -J  mi.  to  integers.  12.  ^  of  a  rod  to  integers. 

5.  ^  bu.  to  integers.  13.  .03125  T.  to  integers. 

6.  -f  A.  to  integers.  14.  -J  yd.  to  integers. 

7.  .1845  gal.  to  integers.  15.  -/%  mi.  to  yards. 

8.  .15625  bu.  to  integers.  16.  -^  mi.  to  rod. 

Reduction  Ascending-. 
1.  Reduce  -f-  gi.  to  the  fraction  of  a  gal. 

Process.  Explanation. 

%  4  gi.  =  1  pt.,  therefore  ^  the  number  of 

§  y  1  \^  1  x  1  __  A  gi.  =  the  number  of  pt.  ;    2  pt.  =  1  qt., 

9         £         2         4         36  therefore  |  the  number  of  pt.  =  the  number 

of  qt.  ;  4  qt.  =  I  gal.,  therefore  i  the  number  of  qt.  =  the  number  of  gal.  ; 
hence  f  gi.  =  f  X  I  X  \  X  i  =  aV  Sal- 


RELATION   OF  ONE  DENOMINATE  TO  ANOTHER     189 


2.  Reduce  .375  wk.  to  the  fraction  of  a  year. 


Process. 
.375  wk.  =  |  wk. 

I  x  7  =     L  da. 
X 


Explanation. 


•375  =  ^VV  =  f  I  wk.  =  I-  of  7  da. 
=  -2J-  da.  365  days  =  1  yr.,  therefore  -2^ 
da.  =  -V-  X  -sh  =  My  yr. 

EXERCISES. 


Reduce : 

1.  -J  gi.  to  gal. 

2.  ^|  min.  to  da. 

3.  ff  ft.  to  mi. 

4.  5f  oz.  to  T. 

5.  f  in.  to  rd. 

6.  2fpt.  to-bu. 

7.  ^  sec.  to  deg. 

8.  3^  min.  to  da. 

9.  .45  Ib.  toT. 

10.  4  cu.  in.  to  cu.  ft. 


11.  £  sq.  yd.  to  A. 

12.  $  pt.  to  bbl. 

13.  f  gi.  to  gal. 

14.  4^  sq.  in.  to  sq.  rd. 

15.  9^  in.  to  mi. 

16.  |  of  |  "l  to  cong. 

17.  5-|  X  7^  cu.  in.  to  cd. 

18.  1  li.  to  mi. 

19.  |f  Ib.  to  T. 

20.  .89725  oz.  to  cwt. 


THE  FRACTIONAL  RELATION  OF  ONE  DE- 
NOMINATE NUMBER  TO  ANOTHER. 

EXERCISES. 

1.  What  part  of  4  ft.  7  in.  is  3  ft.  4  in.  ? 
Process.  Explanation. 

4  ft.  7  in.  =  55  in.  1  in.  =  ^  of  55  in.  ;  therefore,  40  in.  = 

3  ft.  4  in.  =  40  in.         It  of 
.40.  =  _8_ 

2.  What  decimal  part  of  £1  is  16s.  8d. 

1st  Process.  Explanation. 

Both  quantities  must  be   reduced 
to  the  same  denomination. 

£l  =  240d.   16s.  8d.  =  200d.   200d. 

=  m  or  I  of  240d- 


n. 
reduced  =  T8T. 


16s.  8d.  =  200d. 
=  £f  =  £.8333  +. 


. 
reduced  to  a  decimal  =  £.8333  -f-- 


190  PRACTICAL  ARITHMETIC 

3.  What  fraction  of: 

1.  1  yd.  is  2  ft.  9  in.? 

2.  1  mi.  is4rd.  21  yd.? 

3.  1  A.  is  24  sq.  rd.  33  sq.  yd.  ? 

4.  4|  Ib.  is  51  oz.  ? 

5.  3  mi.  is  5  rd.  2  yd.  2  ft.  2  in.? 

6.  3  bbl.  are  13  gal.  3  qt.  3  pt.  3  gi.? 

7.  3  bu.  is  1  bu.  3  pk.  4  qt. 

8.  5  Ib.  troy  is  6  oz.  6  pwt.  6  gr.  ? 

9.  4  Ib.  avoirdupois  is  4  Ib.  troy? 

10.  A  6-in.  cube  is  6  cu.  in.  ? 

11.  65  ch.  is  1430  ft.? 

12.  365  da.  is  4  wk.  4  da.  4  hr.? 

13.  360°  is  40°  40'  ? 

14.  £1  is  18s.  5Jd.? 

15.  1  cwt.  is  16  Ib.  11  oz.? 

4.  What  decimal  fraction  of  1  bu.  is  3  pk.  6  qt.  1  pt.? 

2d  Process.  Explanation. 

1 .  2  pt.  =  1  qt. ;  therefore  £  the  number  of  pt.  =  the 

n  K  number  of  qt.     ^  of  1  —  .5  qt.,  which  added  to  6  qt. 

=  6.5  qt.     8  qt.  =  1  pk.  ;  therefore  1  of  the  number 
of  qt.   =   the   number  of   pk.      \   of  6.5  =   .8125 


.953125  pk.,  which  added  to  3  pk.  ==  3.8125  pk.     4  pk,  = 

1  bu.  ;  therefore  i  the  number  of  pk.  =  the  number 
of  bu.     I  of  3.8125  =  .953125  bu. 

5.  What  decimal  fraction  of: 

1.  1  S.  is  6°  25/36//? 

2.  1  mi.  is  5rd.  3ft.  10  in.? 

3.  1  yd.  is  31  in.  ? 

4.  1  T.  is  3  cwt.  48  Ib.  9  oz.  ? 

5.  1  A.  isl  R.  39  P.? 

6.  1  T.  is  6  cwt.  75  Ib.  ? 

7.  1  da.  is  11  hr.  55  min.  41.7  sec.? 


RELATION  OF  ONE  DENOMINATE  TO  ANOTHER  191 

8.  1  A.  is  4276  sq.  ft.  ? 

9.  1  Ib.  is  14  oz.? 

10.  82  mi.  70  rd.  is  10  mi.  10  rd.? 

11.  228  bu.  3  pk.  is  8  bu.  2  pk.  6  qt? 

12.  1  Ib.  is  4  oz.  8  pvvt.  12  gr.? 

13.  1  mi.  is  765yd.  9  in.? 

14.  1  cd.  is  4  cd.  ft.  8  cu.  ft.  ? 

REVIEW. 

1.  What  will  be  the  cost  of : 

1.  1  T.  15  cwt.  36  Ib.  of  sugar  @  3  cts.  a  pound? 

2.  3  Ib.  9  oz.  13  pwt.  of  gold  dust  @  $.75  a  pwt.? 

3.  8  tons  of  coal  @  $.26£  a  cwt.  ? 

4.  9  barrels  of  flour  at  $.03  a  pound? 

5.  16  Ib.  9  oz.  butter  at  $.30  a  pound? 

6.  4  pk.  5  qt.  cherries  at  10  cts.  a  quart? 

7.  40  rd.  8  ft.  9  in.  fence  at  $.80  per  ft.  ? 

8.  25  bu.  of  seed  at  8  cts.  a  pint  ? 

9.  7  bu.  3  pk.  2  qt.  blackberries  at  7  cts.  a  qt.  ? 

10.  14  hhd.  of  molasses  a{  12  cts.  a  qt.? 

2.  How  many  bu.  of  wheat  in  1260  Ib.  ? 

3.  How  many  min.  in  the  yr.  1898? 

4.  How  many  cords  of  wood  in  a  pile  4  ft.  wide,  7  ft. 
high,  70ft.  long? 

5.  How  many  days  of  12  hrs.  each  will  it  require  to 
make  a  million  figures  if  one  figure  is  made  each  second  ? 

6.  How  many  bu.  of  carrots  will  a  10-acre  field  produce 
if  each  sq.  rd.  produces  5  bu.  ? 

7.  How  many  sec.  are  there  in  365  da.  5  hr.  48  min.  49 
sec.? 

8.  How  many  bu.  of  oats  in  2000  Ib.  ? 

9.  How  many  sec.  from  7  A.M.,  Aug.  15th,  to  Dec.  7th, 
7  P.M.  ? 


192  PRACTICAL  ARITHMETIC 

10.  How  many  kegs,  each  holding  7  gal.  3  qt.  1  pt.,  can 
be  filled  from  11  hhd.  of  wine? 

11.  How   many   degrees   in   a   quadrant   measured   on   a 
meridian  of  the  earth's  surface?     How  many  miles? 

12.  If  1  ton  of  phosphorus  is  used  in  making  10,000,000 
matches,  how  many  gr.  of  phosphorus  on  each  match  ? 

13.  If  a  cistern  holds  4890  gal.  of  water,  how  many  bbl. 
'does  it  hold  ? 

14.  If  hay  at  $15  per  T.  is  exchanged  for  flour  at  $5.85 
per  bbl.,  how  many  bbl.  will  a  ton  of  hay  buy  ? 

15.  If  a  druggist  put  83  43  59  of  a  medicinal  substance 
in  2-gr.  pills,  how  many  pills  did  he  make  ? 

16.  If  a  man  constructed  a  cistern  12  ft.  long  and  8  ft. 
wide  to  hold  150  bbl.,  how  high  did  he  make  it? 

17.  If  10  bales  of  goods  weigh  22  cwt.  86  lb.,  what  will 
155  bales  of  like  size  weigh? 

18.  If  a  silver  dollar  weighs  412J  gr.,  what  will  1,000,000 
dollars  weigh? 

19.  If  a  bbl.  of  flour  costs  £1  4s.  9d.,  how  many  bbl.  can 
be  bought  for  £275  10s.  gd.  ?     (Reduce  before  dividing.) 

20.  If  a  man  travels  24  mi.  7  fur.  30  rd.  in  a  day,  how 
long  will  it  take  him  to  travel  300  mi.  6  fur.  20  rd.  ? 

tion  :  40  rd  =  1  fur. 


21.  If  a  cu.  ft.  of  ice  weighs  58.1  lb.,  how  many  tons  will 
an  ice-house  hold  that  is  45  ft.  long,  32  ft.  wide,  and  20  ft.  high  ? 

22.  Find  the  cost  of  1  qt.  of  olive  oil  when  1  doz.  pt.  cost 
$3.50. 

23.  Find  the  number  of  gal.  in  a  cistern  5J  ft.  square  and 
7  ft.  deep. 

24.  Find  the  cost  of  covering  the  floor  of  a  hall  46J  ft. 
long  and  14  ft.  9  in.  wide  with  matting  1 J  yd.  wide  at  $.25 
a  yard. 


ADDITION  OF  DENOMINATE  NUMBERS  193 

25.  If  a  glacier  moves  uniformly  100  ft.  a  year,  how  far 
will  it  go  in  181  days? 

26.  If  a  man  earns  $3  per  day  and  pays  $6  a  week  for 
board,  etc.,  how  much  can  he  save  in  7  mo.  ? 

27.  A  square  lot,  having  32  chains  on  a  side,  contains  how 
many  acres? 

28.  How  many  times  will  the  wheel  of  a  carriage  17.5  ft. 
in  circumference  revolve  in  going  1  mi.  5  ft.  ? 

29.  How  many  board  ft.  in  3  planks  12  ft.  long,  9  in. 
wide,  and  3J  in.  thick  ? 

30.  What  will  it  cost  to  carpet  a  room  18  ft.  by  24  ft.  with 
carpet  f  yd.  wide  at  $1.25  per  yd.,  the  breadths  to  run  length- 
wise ? 

31.  What  decimal  part  of  a  yr.  has  passed  with  August 
15th? 

ADDITION  OF  DENOMINATE  NUMBERS. 

In  the  addition  of  simple  numbers  we  have  a  uniform 
decimal  scale  ;  in  the  addition  of  compound  numbers  we  have 
a  varying  scale ;  apart  from  this  there  is  no  difference  in  the 

process  of  adding. 

EXERCISES. 

1.  What  is  the  sum  of  12  Ib.  5  oz.  13  pwt,  21  Ib.  8  oz.  15 
pwt.,  13  Ib.  7  oz.  10  pwt.,  51  Ib.  3  oz.  17  pwt.  ? 

Process.  Explanation. 

12         20  Units  of  the  same  denomination  must  stand  in 

Ib.       oz.      pwt.  the  same  column. 

12        5        13  The  scale  is  24,  20,  12.     We  use  20  and  12. 

£)1  1  p.  The  sum  of  the  pwt.  is  55.     55  pwt.  =  2  oz  15 

pwt.     We  write  the  15  under  the  column  of  pwt. 

I '^         •  and  add  the  2  oz.  with  the  column  of  ounces.     The 

51         3         17  sum  of  the  oz.  is  25,  which  equals  2  Ib.  and  1  oz. 

QQ         i         15  We  write  the  1  oz.  under  the  column  of  oz.  and  add 

the  2  Ib.  with  the  column  of  Ib.,  making  99  Ib. 

13 


194 


PRACTICAL   ARITHMETIC 


2.  What  is  the  sum  of  37  A.  159  P.  25  sq.  rd.  8  sq.  ft. 
126  sq.  in.,  20  A.  110  P.  30  sq.  rd.  8  sq.  ft.  131  sq.  in.,  345 

Process.  A.  Ill  P.  16  sq.  rd.  7 

sq.  ft.  99  sq.  in? 

Explanation. 

\  sq.  yd.  —  4|  sq.  ft. 

\  sq.  ft.  =  72  sq.  in. 
Adding  4  sq.  ft.  and  72 
sq.   in.,   we  have   a  result 
free  from  fractions. 

3.  Find  the  sum  of  the  following : 

PO 

rd.         yd.      ft.        in. 

140  5  2  7 
225  0  3  9 
402  4  0  10 


160 

30£ 

9 

144 

A. 

P. 

sq.  rd. 

sq.  ft. 

sq.  in. 

37 

159 

25 

8 

126 

20 

110 

30 

8 

131 

345 

111 

16 

7 

99 

404 

62 

12(1) 

=  7 

68 

i 

==   ^(V 

1  =  72 

404 

62 

13 

2 

140 

Suggestion  :  Reduce 

the  i 

yd 

.  occurring  in  the 

result  to  feet 

and  inches. 

(2.) 

(3.) 

mi.      rd. 

yd. 

ft. 

in. 

A. 

P.    sq.  yd.    sq 

.ft. 

sq.  in. 

5     251 

4 

2 

9 

112 

80 

21 

5 

0 

5     184 

4 

0 

6 

108 

75 

16 

4 

0 

8     256 

5 

1 

7 

93 

57 

12 

0 

0 

7     159 

4 

0 

8 

115 

18 

28 

0 

0 

(4.)                                                 (5.) 

Ib. 

oz. 

pwt. 

gr. 

T. 

cwt. 

Ib. 

oz. 

15 

9 

17 

11 

4 

6 

38 

9 

14 

8 

16 

23 

9 

12 

49 

12 

15 

6 

3 

18 

14 

4 

44 

11 

12 

10 

0 

19 

9 

20 

10 

24 

3 

16 

5 

21 

5 

12 

8 

13 

0 

14 

0 

7 

9 

65 

6 

ADDITION  OF  DENOMINATE  NUMBERS  195 

6.  6  mi.  80  rd.  3  yd.  2  ft.  1  in.,  4  mi.  75  rd.  1  yd. 
2  ft.  7  in.,  5  mi.  170  rd.  2  yd.  1  ft.  8  in. 

4.  Find  the  value  of  £  mi.  +  13-J-  rd. 

Process. 

f  mi.  =  266  rd.  3  yd.  2  ft. 
13J-  rd.  =    13  rd.  1  yd.  2  ft.  6  in. 
Sum       ==  279  rd.  5  yd.  1  ft.  6  in. 

Explanation. 

5  yd.  1  ft.  6  in.  =  16J  ft.  =  1  rd.     279  rd.  +  1  rd.  =  280  rd. 

5.  Find  the  value  of: 

1.  |  mi.  +  .46  rd.  -f  3|  rd. 

2.  .oo|  sq-  yd-  +  -°4f  sq- ft-  +  -0008  sq- in- 

3.  £f  -f  3.75s.  +  .975d. 

4.  .2965  T.  +  .8725  cwt.  +  .3725  cwt.  +  .1625  Ib. 

5.  |  Ib.  +  3f  oz.  +  5|  pwt. 
6-  -h  7r-  +  A  wk.  +  -ft-  hr. 
7.  A  mi.  +  |  rd.  +  |  yd. 

8.  W°  +  «'  +  «". 

9.  £|  +  |  of  5f  s. 

10.  f  wk.  +  f  hr.  +  ft-  min. 

11.  |  A.  +  $  sq.  rd.  -f  f  sq.  yd. 

12.  274-  cwt.  +  26£  Ib.  -f  14  oz.     [112  Ib.  =  1  cwt.] 

13.  1|  hhd.  -f  36  gal.  3  qt.  1 J  pt.  +  |  gal.  +  2  qt. 

f  pt.  +  1.75  pt. 

14.  f  of  £13  +  i  of  -/T  of  f  of  £2  12s.  +  f  of  9d. 

6.  Add  |  of  |  of  a  guinea  to  .4  of  .375  of  £1,  and  express 
the  sum  as  the  decimal  of  a  crown  (5s.). 

7.  Express  .05735  mi.  -f-  46.25  yd.  as  the  decimal  of  7  fur. 

8.  What  is  the  value  of  1.1375  fathoms  +   .875  yd.  -f 
2.965  ft.  +  9.75  in.  in  feet. 


196  PRACTICAL   ARITHMETIC 

SUBTRACTION   OF   DENOMINATE  NUMBERS. 

EXERCISES. 

1.  From  2  mi.  116  rd.  4  yd.  0  ft.  4  in.  take  1  mi.  120  rd. 
2  yd.  1  ft.  8  in. 

Process.  Explanation. 

mi.      rd.      yd.    ft.    in.  Units  of  the  same  denomination  must  stand 

2      116      404  *n  ^e  same  column.     Since  we  cannot  sub- 

tract  8  in'  from  4  in'    we  ac^  to  the  ®  ft'  °ne 


1  20      2      1       8 

of  the  4  yds.  ;   1  yd.  =3  ft.  ;  now  having  3 


316      118  ft  ^  instead  of  0  ft.,  we  add  to  the  4  in.  one  of 

the  3  ft.  ;  1  ft.=  12  in.  ;  12  in.  -f  4  in.  =  16  in.  ;  16  in.  —  8  in.  =  8  in. 
Proceeding  to  the  feet,  we  say,  "1  ft.  from  2  ft.  leaves  1  ft."  Proceeding 
to  the  yards,  we  say,  "2  yd.  from  3  yd.  leaves  1  yd."  One  of  the  2  mi. 
added  to  116  rd.  gives  us  320  +  116  =  436  ;  436  rd.  —  120  rd.  =  316  rd. 


(2- 

) 

(3.) 

£ 

s. 

d. 

A. 

sq.  rd. 

sq.  yd. 

sq.  ft. 

sq.  in. 

From 

37 

17 

9 

From 

18 

40 

25 

6 

100 

take 

29 

18 

10 

take 

9 

50 

13 

7 

140 

(4.)  (5.) 

T.    cwt.     Ib.      oz.  Ib.      oz.    pwt.    gr. 

From  5     13     21     13         From  284     0       0       0 
take  3     19       2     14  take  100     9     17     21 

(6-)  (7.) 

yr.      wk.    da.     hr.     min.    sec.  S.       °         /         // 

From  99  36  5  31  46  49     From  12  25  20  43| 
take  81  46  6  32  47  50      take  10  28  49  57f 

8.  From  1  hhd.  38  gal.  3  qt.  2  pt.  take  60  gal.  2  qt.  1  gi. 

9.  From  8  Ib.  take  1  Ib.  \l  23  29. 

10.  From  5  T.  take  10  Ib.  8  oz. 

11.  From  -f  oz.  take  -J  pwt. 


SUBTRACTION  OF  DENOMINATE  NUMBERS    197 

Process.  Explanation.. 

3.  oz>      —  7  pwt.  12  ST.  Reduce  the  fractions  to  lower  denom- 

Y          ,    _  01  inations  and  then  subtract. 

6  pwt.  15  gr. 

12.  From  2£  oz.  take  |  pwt. 

13.  From  ^-  da.  take  -|  min. 

14.  From  -^  hhd.  take  f  qt. 

15.  From  \  wk.  take  .9  da. 

16.  From  f  pk.  take  .0625  bu. 

17.  From  .625  Troy  Ib.  take  4.25  Troy  oz. 

18.  From  ^  sq.  rd.  take  f  sq.  yd. 

19.  From  45  sq.  yd.  take  45  sq.  in. 

20.  From  360°  take  T4r  of  a  quadrant. 

21.  Find  the  lapse  of  time  between  July  4,   1890,  and 
August  15,  1898. 

Process.  Explanation. 

1898      8      15  JulJ  Js  ^e  7th  month  and  Aug.  the  8th  month 

1890     7       4         of  the  calen^ar- 
8     1     11 

22.  Between  Jan.  9,  1842,  and  Mar.  4,  1898. 

Process.  Explanation. 

1898      3      4  January   is   the    1st    month    and   March    the    3d 

1842      1       9  month  of  the  calendar.     1  month  =  30  days  in  most 


56     1  25 


computations. 


23.  Between  Mar.  2,  1857,  and  July  4,  1866. 

24.  Between  Jan.  5,  1844,  and  Mar.  16,  1862. 

25.  Between  May  3,  1804,  and  Dec.  16,  1871. 

26.  The   Spanish- American   war   began   April   21,   1898, 
and  ended  Aug.  12,  1898.     Find  the  difference  of  the  dates. 

27.  The  American  civil  war  began  April  11,  1861,  and 
ended  April  9,  1865.     How  long  did  it  continue? 


198  PRACTICAL  ARITHMETIC 

28.  The  Revolution  commenced  April  19,  1775,  and  closed 
Jan.  20,  1783.     How  long  did  the  war  last? 

29.  Columbus  discovered  America  Oct.  11,  1492.     How 
long  ago  did  that  event  occur? 

30.  A  note  dated  Aug.  10.,  1882,  was  paid  Nov.  11,  1887. 
How  long  did  it  run  unpaid  ? 


MULTIPLICATION  OF  DENOMINATE  NUMBERS. 

EXERCISES. 
1.  Multiply  5  gal.  3  qt.  1  pt.  3  gi.  by  9. 

Process.  Explanation. 

gal.  qt.    pt.    gi.  9  times  3  gi.  =  27  gi.  =  6  pt.  3  gi.    We  reserve 

5313  the  6  pt.  to  add  to  the  next  product.     9  times  1  pt. 

=  9  pt.  ;  9  pt.  -f  6  pt.  reserved  =  15  pt.  =  7  qt. 
and  1  pt.     We  reserve  the  7  qt.     9  times  3  qt.  = 


53      2      1      3  27  qt      27  ^    ^_  7  qt    reserved  _  34  qt   _  8  gaj 

and  2  qt.     We  reserve  the  8  gal.     9  times  5  gal.  =  45  gal. ;  45  gal.  +  8 
gal.  reserved  =  53  gal. 

2.  Multiply  18  Ib.  9  oz.  4  pwt.  16  gr.  by  11. 

3.  Multiply  2  T.  2  cwt.  46  Ib.  7  oz.  by  8. 

4.  Multiply  9  mi.  3  fur.  20  rd.  3  yd.  2  ft.  by  6. 

5.  Multiply  6  yr.  5  mo.  15  da.  18  hr.  by  12. 

6.  Multiply  26  cd.  3  cd.  ft.  12  cu.  in.  by  18. 

7.  Multiply  £9  17s.  6d.  1  far.  by  28. 

8.  Multiply  5  T.  8  cwt.  64  Ib.  8  oz.  by  37. 

9.  Multiply  7  mi.  4  fur.  15  rd.  3  yd.  2  ft.  8  in.  by  48. 

10.  Multiply  5  Ib.  7  oz.  15  pwt.  19  gr.  by  75. 

11.  Multiply  21  Ib.  93  23  19  16  gr.  by  25. 

12.  Multiply  9  A.  3  R.  22  P.  6  sq.  yd.  5  sq.  ft.  by  10. 

13.  Multiply  25°  37'  51"  by  16. 

14.  Multiply  £10  18s.  7d.  2  far.  by  29. 

15.  Multiply  8  mi.  120  rd.  4  yd.  by  26. 


DIVISION  OF  DENOMINATE  NUMBER  199 

DIVISION  OF  DENOMINATE  NUMBERS. 

EXERCISES. 
i.  Divide  76  Ib.  10  oz.  14  pwt.  12  gr.  by  6. 

Process.  Explanation. 

Ib.      oz.     pwt.     gr.  To  divide  a  quantity  by  6  is  to  take  ^ 

6)  76      10      14      12  of  it;-     i  of  7(>  lb-  =  12  lb-  and  4  lb-  re' 

~Vo         q       TK      -To~  maining ;  4  lb.  =  48  oz.  ;  48  oz.  -f-  10  oz. 

=  58  oz.     i  of  58  oz.  =  9  oz.  and  4  oz. 

remaining ;  4  oz.  —  80  pwt.  ;  80  pwt.  -f  14  pwt.  =  94  pwt.  ;  $  of  94  pwt. 
=  15  pwt.  and  4  pwt.  remaining  ;  4  pwt.  =  9G  gr. ;  96  gr.  -{-  12  gr.  =  108 
gr.  ;  i  of  108  gr.  =  18.  Hence  the  quotient  is  12  lb.  9  oz.  15  pwt  18  gr. 

2r  Divide: 

1.  112T.  16  cwt.  66  lb.  by  7. 

2.  17  bu.  3  pk.  4  qt.  by  8. 

3.  29  lb.  53  33  19  by  9. 

4.  125S.  24°  12'  by  10. 

5.  427  A.  131  sq.  rd.  by  11. 

6.  342  gal.  2  qt,  1  pt.  2  gi.  by  5. 

7.  16  T.  1300  lb.  by  12. 

8.  120  mi.  313  rd.  3  yd.  2  ft.  by  12. 

9.  £31  5s.  8d.  by  4. 

10.  196  cd.  4  cd.  ft.  12  cu.  ft.  by  36. 

11.  £275  10s.  6d.  by  £1  4s.  9d. 

Process. 
£275  10s.  6d.  =  66,1 26d. 

£1  4s.  9d.    =  297d. 
66,1 26d.  -~  297d.  =±±  222ff. 

What  rule  is  derivable  from  the  process? 

12.  48  T.  9  cwt.  23  lb.  8  oz.  by  6  T.  1  cwt.  1 5  lb.  7  oz. 

13.  200  mi.  6  fur.  18  rd.  by  24  mi.  7  fur.  22  rd. 

14.  31  cwt.  18  lb.  by  3  lb.  8  oz. 


200  PRACTICAL   ARITHMETIC 

15.  13  Ib.  7  oz.  15  pwt,  by  2  oz.  10  pwt. 

16.  5f  mi.  by  7  ft.  4  in. 

17.  118  bu.  2  pk.  by  7  bu.  1  pk.  5  qt. 

18.  35  wk.  3  da.  15  hr.  25  min.  by  17  wk.  6  da.  22  hr. 

39  min. 

19.  61  ft.  3  in.  by  8  ft.  7  in. 

20.  A  quadrant  by  27°  14'  45". 

LONGITUDE  AND  TIME. 

INDUCTIVE   STEPS. 

1.  Does  the  earth  revolve  on  its  axis  from  west  to  east  or 
from  east  to  west  ? 

2 .  It  revolves  once  in  how  many  hours  ? 

3.  Does  the  sun  actually  revolve,  or  only  appear  to  revolve 
around  the  earth  ? 

4.  If  the  earth  revolves  from  west  to  east,  do  Eastern  or 
Western  people  behold  the  sun  first  ? 

5.  Has  a  place  30°  east  of  Philadelphia  later  or  earlier 
time? 

6.  When  it  is  noon  in  New  York,  is  it  afternoon  or  fore- 
noon in  Chicago? 

7.  Through  how  many  degrees  does  the  sun  appear  to  travel 
in  24  hrs.  ? 

8.  Then  how  many  degrees  of  longitude  and  how  many 
hours  are  compassed  in  a  day  ? 

Time.        Longitude. 
24  hrs.     =     360°. 
Dividing  the  equation  by  24,  we  have : 

1  hr.       =     15°. 
Dividing  by  60,  we  have : 

1  min.    =     15'. 
Dividing  again  by  60,  we  have : 

1  sec.     =     15". 


LONGITUDE  AND  TIME  201 

RULE. 

To  reduce  time  to  longitude,  multiply  by  15  ;  to  reduce 
longitude  to  time,  divide  by  15. 

PROBLEMS. 

1.  The  difference  of  time  between  two  places  is  1  hr.  15 
min.  30  sec.     Find  the  difference  of  longitude. 

Process.  Explanation. 

hr.     min.     sec.  Since  we  are  required  to  reduce  time  to  longi- 

1         15        30  tude,  we  multiply  the  given  hours,  minutes,  and 

seconds  by  15,  and  obtain  18°  52'  20". 


18°     S2      30"  2.  The  difference  of  longitude  between 

New   York   and   Baltimore    is    2°    36'. 
Find  the  difference  of  time. 

Process.  Explanation. 

15)2°  36'  Since  we  are  required  to  reduce  lon- 

10  m[u    24  Sec  gitude    to    time,    we    divide    the    given 

number  of  degrees  and  minutes  by  15, 
and  obtain  10  min.  24  sec.  as  the  difference  of  time. 

The  Meridian  of  a  place  is  an  imaginary  line  running  from 
North  Pole  to  South  Pole  through  that  place. 

A  meridian  divides  longitude  into  east  longitude  and  west 
longitude,  making  180°  of  each. 

The  meridian  of  Greenwich,  near  London,  or  of  Washing- 
ton, D.  C.,  is  commonly  reckoned  from. 

3.  The  longitude  of  Washington  (from  Greenwich)  is  77° 
2'  48"  W.,  and  of  San  Francisco  122°  24'  15"  W.     Find  the 
difference  of  longitude  and  the  difference  of  time. 

4.  If  the  difference  of  time  between  San  Francisco  and 
Philadelphia  is  3  hr.  9  min.  7  sec.,  what  is  the  longitude  of 
Philadelphia? 


202  PEACTICAL   ARITHMETIC 

5.  The  difference  in  time  between  Berlin  and  New  York 
is    5    hr.   49    min.   35   sec.      Find   the   difference   in    longi- 
tude. 

6.  If  Berlin  is  13°  23'  43"  E.,  find  how  much  of  the 
preceding  difference  is  west  longitude. 

7.  In  travelling  west  my  watch  seemed  to  gain  20  min. 
How  many  degrees  did  I  travel? 

8.  Constantinople  is   28°  59'  E.     When  it  is  noon   in 
Greenwich,  what  is  the  time  in  Constantinople? 

9.  When  one  place  is  in  west  longitude  and  the  other  in 
east  longitude,  do  you  add  or  subtract  to  find  the  difference 
of  longitude? 

10.  New  York  is  74°  3'  west;  Paris,  France,  is  2°  20' 
east.     Find  the  difference  of  time. 

11.  When  it  is  noon  at  Boston  (71°  3'  30"  west),  what  is 
the  time  at  Paris  (2°  20'  22"  east)? 

12.  Canton  in  China  is  113°  14'  30"  east  longitude  and 
Washington  is  77°  west  longitude.     When  it  is  midnight  on 
July  4th,  at  Washington,  what  time  will  it  be  at  Canton? 

Standard  Time. 

Nov.  18,  1883,  the  United  States  was  divided  into  four 
time-belts,  each  15°  wide,  named  respectively,  Eastern,  Central, 
Mountain,  and  Pacific.  The  local  time  of  the  middle  meridian 
of  each  time-belt  was  adopted  as  the  standard  time  of  the 
whole  belt. 

1.  Eastern  time  is  that  of  the  75th  meridian. 

2.  Central  time  is  that  of  the  90th  meridian. 

3.  Mountain  time  is  that  of  the  105th  meridian. 

4.  Pacific  time  is  that  of  the  120th  meridian. 

All  places  lying  within  7°  30'  of  the  middle  meridian  have 
the  time  of  that  meridian. 


204  PRACTICAL   ARITHMETIC 

PROBLEMS. 

1.  St.  Paul  is  in  longitude  93°  5'  W.     Find  the  difference 
between  the  local  and  the  standard  time  of  St.  Paul. 

Explanation. 

St.  Paul  is  within  7°  30'  of  90°,  and 
is  within  the  central  time-belt.  Its  dis- 
tance from  the  90th  meridian  is  3°  5', 


oo    c/ 

which,  divided  by  15,  gives  12  min.  20 

12  mm.  20  Sec.  sec.,  the  difference  of  time  required. 

2.  Boston  is  in  longitude  71°  3'  30".     Find  the  difference 
between  the  local  and  the  standard  time  of  Boston. 

3.  Pittsburg  is  within  7°  30'  from  Philadelphia,  which  lies 
close  to  the  nliddle  meridian  of  the  eastern  belt.     When  it  is 
noon  at  Philadelphia,  what  is  the  standard  time  at  Pittsburg  ? 

4.  Galveston  is  in  longitude  94°  50'  W.     When  it  is  noon 
there  by  local  time,  what  hour  is  it  by  standard  time  ? 

5.  St.  Louis  is  in  longitude  90°  15'  15"  W.     When  it  is 
noon  there  by  standard  time,  what  is  the  local  time? 

MISCELLANEOUS   PROBLEMS. 

1.  If  one  doz.  pints  of  oil  cost  $4.00,  what  is  the  cost  of 
one  qt.  ? 

2.  A  gentleman  in  travelling  found  at  a  certain  railroad 
station  that  his  watch  was  1  hr.  and  25  min.  slow.     What 
direction  was  he  travelling?     How  far  had  he  travelled? 

3.  A  note  dated  June  12,  1896,  was  paid  Jan.  5,  1897. 
How  long  did  the  note  run  ? 

4.  How  many  steps  |-  yd.  long  will  a  man  take  in  walk- 
ing 1  mi.  and  580  yd.  ? 

5.  If  10  grain  bins  contain  254  bu.  3  pk.  7  qt.   1  pt., 
what  does  1  bin  contain? 

6.  Since  noon  the  sun  has  seemed  to  pass  through  10° 
43'  35".     What  is  the  time  of  day  ? 


MISCELLANEOUS  PROBLEMS  205 

7.  If  a  cu.  ft.  of  water  weighs  1000  oz.,  how  many  Ib. 
avoirdupois  does  a  cu.  yd.  of  water  weigh  ? 

8.  When  it  is  1  hr.  37  min.  12  sec.  P.M.  at  Bangor  (68° 
47'  W.),  what  is  the  time  at  St.  Paul  (93°  5'  W.)  ? 

9.  A  crib  measuring  16  ft.  X  6  ft.  9  in.  X  7  ft.  is  full  of 
corn  in  the  ear.    How  many  bu.  of  shelled  corn  will  there  be  ? 

10.  In  556,688  ft.  how  many  miles? 

11.  How  many  gal.  of  air  in  a  room  16  ft.  long,  11  ft. 
wide,  and  10ft.  high? 

12.  How  many  bu.  of  shelled  corn  will  fill  a  vat  that  holds 
6000  gal.  of  water? 

13.  A  block  of  marble  4  ft.  long  and  2J  ft.  wide  contains 
12J  cu.  ft.     How  thick  is  the  block  ? 

14.  How  many  bu.  in  6  tons  of  oats  ? 

15.  How  much  is  gained  on  65  doz.  eggs  bought  at  $.15  a 
doz.  and  sold  at  the  rate  of  1^  doz.  for  $.25  ? 

16.  What  is  the  cost  of  4  tons  and  468  pounds  of  hay  at 
$12  a  ton? 

17.  A  firkin  of  butter  weighed  61  Ib.  12  oz.     How  much 
did  the  vessel  itself  weigh  ? 

18.  If  a  man  can  do  a  piece  of  work  in  22  hr.  30  min.  25 
sec.,  what  part  of  it  can  he  do  in  13  hr.  11  min.  15  sec.? 

19.  Divide  3  gal.  2  qt.  2.03  pt,  by  18,  and  reduce  the  result 
to  the  decimal  of  a  barrel. 

20.  What  decimal  of  a  Ib.  avoirdupois  is  a  Ib.  troy  ? 

21.  How  many  bu.  of  potatoes  in  2240  Ib? 

22.  The  longitude  of  New  York  is  74°  0'  3"  W. ;   of 
London,  0°  5'  48"  W.     Find  the  difference  of  time  between 
the  two  cities.     Which  has  the  earlier  time  ? 

23.  If  a  bicycle  wheel  7  ft.  4  in.  in  circumference  makes  3 
revolutions  in  a  second,  at  what  rate  per  hour  is  the  rider  going  ? 

24.  How  many  francs  equal  $1.00? 

25.  Reduce  £3  8s.  4d.  to  dollars,  U.  S.  currency. 


206 


PRACTICAL   ARITHMETIC 


26.  The  annual  cost  of  Spanish  royalty  is  9,500,000  pesetas. 
Reduce  to  U.  S.  money.     (Peseta  =  $.193.) 

27.  Latitude  is  distance  north  or  south  from  the  Equator. 
If  a  man  travels  due  north  from  the  Equator  2500  mi.,  what 
latitude  does  he  reach  ?     (1°  =  69£  mile.) 

28.  3780  gal.  of  water  will  fill  how  many  barrels  ? 

29.  If  hyoscine  hydrobromate  is  worth  $12.50  a   grain, 
what  will  be  the  cost  of  12  tablets  of  the  drug,  each  con- 
taining .01  of  a  grain  ? 

30.  A  owns  -fj-  of  a  farm,  and  B  owns  the  remainder.     |  of 
the  difference  between  their  shares  is  16  A.  80  sq.  rd.    Find  the 
share  of  each  in  acres. 


REVIEW. 
Define : 

1.  Denominate  Number.  19. 

2.  Compound   Denomi-  20. 

nate  Number.  21. 

3.  Money.  22. 

4.  U.  S.  Money.  23. 

5.  Sterling  Money.  24. 

6.  Reduction.  25. 

7.  Reduction  Descending.  26. 

8.  Reduction  Ascending.  27. 

9.  Extension.  28. 

10.  Linear  Measures.  29. 

11.  Surface  Measures.  30. 

12.  Measures  of  Volume.  31. 

13.  Measures  of  Capacity.  32. 

14.  Angle.  33. 

15.  Rectangle.  34. 

16.  Square.  35. 

17.  Area.  36. 

18.  Solid,  37. 


Rectangular  Solid. 

Cube. 

Volume. 

Solid  Contents. 

Board  Measure. 

Weight. 

Troy  Weight. 

Apothecaries'  Weight. 

Circular  Measure. 

Circle. 

Circumference. 

Arc. 

Quadrant. 

Degree. 

Fractional  Relation. 

Uniform  Scale. 

Varying  Scale. 

Longitude. 

Standard  Time. 


REVIEW 


207 


2.  Repeat  the  table  of : 

1.  U.  S.  Money. 

2.  English  Money. 

3.  French  Money. 

4.  Linear  Measure. 

5.  Surveyors'  Linear 

Measure. 

6.  Surveyors'  Square 

Measure. 

7.  Liquid  Measure. 

8.  Apothecaries'  Liquid 

Measure. 

3.  Name  the : 

1.  U.  S.  Coins. 

4.  Repeat  the  rule  for  : 

1.  Reduction  Descending. 

2.  Reduction  Ascending. 

3.  Area  of  Rectangle. 

4.  Volume. 

5.  What  is  the  unit  of: 

1.  U.S.  Money? 

2.  Canadian  Money? 

6.  What  is  the  unit  for  : 

1.  Land? 

2.  Plastering? 

3.  Paving? 

4.  Roofing,  etc.  ? 


9.  Dry  Measure. 

10.  Avoirdupois  Weight. 

11.  Troy  Weight. 

12.  Apothecaries'  Weight. 

13.  Time. 

14.  Months  (Stanza). 

15.  Circular  Measure. 

16.  Counting. 

17.  Paper. 

18.  Books. 


2.  English  Coins. 

5.  Time  to  Longitude. 

6.  Longitude  to  Time. 

7.  Board  Measure. 

8.  Fractional  Relation. 

3.  French  Money  ? 

4.  English  Money  ? 

5.  Bricklaying,  etc.  ? 

6.  Excavations,  etc.? 

7.  Brickwork? 

8.  Grain? 


PART    II. 


PERCENTAGE. 

INDUCTIVE  STEPS. 

1.  A  man  earned  $5  and  spent  $1.00.     What  fractional 
part  of  the  $5  did  he  spend?     What  part  of  $10  would  he 
have  spent  ?     What  part  of  $50  ?     What  part  of  100  ? 

2.  $20  out  of  $100  means  20  per  hundred,  or  20  per  cent. 

3.  What  is  the  meaning  of  10  per  cent?     Of  25  per  cent.? 
Of  50  per  cent.  ?     Of  75  per  cent.  ?     Of  100  per  cent.  ? 

4.  Having  taken  100  per  cent,  of  a  sum  of  money,  how 
much  is  left? 

5.  How  much  is  1  per  cent,  of  $100?   Of  $200?   Of  $1000? 

6.  What  is  5  per  cent,  of  $200?     Of  100  acres?     Of  500 
men? 

7.  What  is  6  per  cent,  of  $600?     Of  900  sheep?     Of 

1200  yards? 

DEFINITIONS. 

1.  Percentage  means  computation  by  the  hundred,  and  has 
100  for  its  unit.     One  per  cent,  of  any  number  is  y^  of  it; 
5  per  cent,  is  y^  of  it. 

Per  cent,  is  a  contraction  of  the  Latin  per  centum,  by  the  hundred. 

2.  The  result  of  computation  is  also  called  Percentage. 

Ts^  Of  $1000  =  $50,  the  percentage. 

3.  The  Symbol  for  per  cent,  is  %.     Per  cent.,  however, 
may  be  expressed  in  five  different  ways  :  6  per  cent.  =  6  %  = 
.06  =  yf^  =  -^j-.     The  best  way  in  any  given  case  is  the  one 
that  affords  the  shortest  solution. 

208 


PERCENTAGE  209 

4.  The  Rate  per  cent,  is  the  number  of  hundredths  ;  T|~g- 
indicates  that  the  rate  is  5  per  cent. 

5.  The  number  on  which  the  percentage  is  computed  is-  the 
Base.     Attention,  therefore,  must  be  given  to  Base,  Rate,  and 
Percentage. 

6.  Amount  is  the  Base  plus  the  Percentage. 

7.  Difference  is  the  Base  minus  the  Percentage. 

EXERCISES. 

1.  Use  10,  100,  16f,  125,  J,  .00625,  in  five  different  ways 
to  express  rate  per  cent. 

Process. 

10  per  cent.  =  10%  =  .10  =  ^  a  =  TV 

100  per  cent,  =  100%  =  1.00  =  {%$  =  1. 
16|  per  cent.  =  16f  %  =  .16f  =  ffl  =  Gfifr)  =  f 
125  per  cent.  =  125%  =  1.25  =  ifj>-  =  f. 
|  per  cent.  =  1%  =  .005  =  ^  =  2W- 
.00625  per  cent.  =  .00625%  -  yiftftWinF  =  TGTO^ 

2.  In  like  manner  express  as  rate  per  cent,  the  following 
numbers  :  15,  20,  25,  40,  50,  55,  65,  75,  96,  45,  85,  121 

8|,  61  66|,  871  371.,  Hi  18|,  621  250,  375,  fc  1   ^, 
ft,  1  .3,  .06,  .0121  .001. 

3.  Change  ^,  1,1  J,  ^  ^^  into  the  symbol  form. 


Process. 


=  W  =  100%- 

=  m  =  10  = 


=  ^W*-  =  -00625%. 
4.  Change  the  following  fractions  into  th«  symbol  form 
IT>  rV  i  i  i>  i  ro>  i>  |»  ft  i  1.  1,  ft  xV  f  »  ft  ft  A- 


14 


210 


PRACTICAL    ARITHMETIC 


5.  Give  the  following  symbol  forms  their  simplest  frac- 
tional form:  90%,  871%,  80%,  75%,  70%,  66f%,  621%, 
60%,  50%,  40%,  371%,  33^%,  30%,  25%,  20%,  16f%, 


To  Find  the  Percentage. 
EXERCISES. 

1.  What  is  25%  of  $24.00? 

Process.  Explanation. 

25%  =  £.      1  of  $24.00  =  $6.00.  The  base  is  $24.00;   the 

Or   <fi>24  00  X    25  —  Hlfi  00  ™te  is  26&  ;  th°  PercentaSe 

Jr,  ^4.1K.  ^b.UU.  is  required     Since  25^  ^ 

|,  25^  of  $24.00  =  £  of  124.00  =  $6.00,  the  percentage. 
Hence  the  formula  : 

Percentage  =  Base  X  Bate. 

NOTE.  —  As  an  important  preparation,  let  the  pupil  write  in  a  table  the 
following  rates,  both  as  common  and  decimal  fractions,  and  make  solution 
with  both  forms. 

2.  What  is: 


1.  70%  of  30  sheep? 

2.  331%  of  9  books? 

3.  28|%  of  35ft? 

4.  121%  of  48  A.? 

5.  50%  of  600yd.? 

6.  621%  of  64  da.? 

7.  4%  of  200  gal.  ? 

8.  80%  of  60  horses? 

9.  25%  of  124yd.? 

10.  5%  of  700  men? 

11.  4%  of  1000  horses? 

12.  40%  of  800  lb.? 

13.  371%  of  160  oxen? 

14.  331%  of  $900? 

15.  15%  of  $500? 


16.  20%  of  400  bu.? 

17.  70%  of  500  yr.? 

18.  90%  of  9000  sec.? 

19.  871%  of  1600  books? 

20.  80%  of  1000  horses? 

21.  75%  of  1000  oz.? 

22.  66f%  of  1500  gal.? 

23.  621%  of  $2400? 

24.  30%  of  10,000  fr.? 

25.  16f%  of  g^  of  abbl.? 

26.  121%  Of  4  of  a  yr.  ? 

27.  10%  of  a  million? 

28.  81%  of  1728  cu.  in.? 

29.  61%  of  144sq.  in.? 

30.  |%  of  $2856.00? 


PERCENTAGE  211 

PROBLEMS. 

1.  Of  600  trees,  33-J%  are  peach  trees.     Find  the  number 
of  peach  trees? 

2.  A  bicycle  marked  $90  was  sold  at  a  reduction  of  12-|-%. 
Find  the  reduction  and  the  selling  price. 

3.  If  a  man  owes  $3564  and  pays  30%  of  it,  how  many 
dollars  does  he  pay? 

4.  Mr.  A.  deposited  in  bank  $963,  and  afterwards  drew 
out  5%  of  it.     How  many  dollars  did  he  draw  out? 

5.  A  farmer  who  had  a  flock  of  540  sheep,  sold  33-^%  of 
them.     How  many  did  he  sell,  and  how  many  had  he  left? 

6.  A  merchant  bought  goods  for  $630,  and  sold  them  at  a 
gain  of  23%.     How  much  did  he  gain? 

7.  How  much  is  made  by  selling  at  20%  profit  a  house 
which  cost  $10,500  ? 

8.  How  much  is  lost  by  selling  at  8%  below  cost  163 
tons  of  coal  which  cost  $6.00  per  ton. 

9.  I  sold  a  horse  which  cost  me  $250  at  a  loss  of  35%. 
What  did  I  get  for  him? 

Is  the  difference  asked  for? 

10.  A  merchant  paid  $.80  a  yard  for  silk.     For  how  much 
must  he  sell  it  to  gain 


Is  the  amount  to  be  sought? 

11.  I  bought  a  bill  of  goods  amounting  to  $986.60,  from 
which  was  deducted  5%.     Find  the  percentage  allowed  and 
the  amount  paid. 

12.  A  certain  mine  yields  60%  of  metal,  and  of  the  metal 
|%  is  silver.     Find  how  much  silver  and  how  much  other 
metal  are  obtained  from  1300  tons  of  ore. 

13.  In  a  school  of  80  children  17J%  are  girls.     Find  the 
number  of  boys. 


212  PRACTICAL   ARITHMETIC 

14.  Assuming  that  gunpowder  contains  75%  of  saltpetre, 
10%  of  sulphur,  15%  of  charcoal,  find  how  many  pounds  of 
each  there  are  in  a  ton  of  powder. 

15.  Express  as  a  rate  per  cent.  .33333  J  and  apply  it  to 
99,999  as  a  base.     Change  .33  J%  to  a  common  fraction  in  its 
lowest  terms,  and  apply  it  to  99,900  as  a  base. 

To  Find  the  Rate. 

Since    Percentage    is    the    product    of    Base    and    Kate, 

obviously 

Rate  =  Percentage 
Base 

EXERCISES. 
1.  What  rate  per  cent,  of  $276  is  $82.80? 

Process.  By  Analysis. 

276  =  100%  of  B. 


82.80  —  ^M  =  30%  of  B. 
Explanation. 

The  base  is  $276  ;  the  percentage  is  $82.80  ;  the  rate  is  required.    Since 

p 

the  rate  is  required,  we  use  the  formula  R.  =  --,  and  obtain  R.  =  30$. 

B. 

2.  What  per  cent,  of: 

1.  $450  is  $90?  11.  812  T.  is  203  T.? 

2.  $12  is  15  cents?  12.  $5600  is  $1600? 

3.  15  Ib.  is  5  Ib.  10  oz.?  13.  64%  is  5J%  ? 

4.  250  head  of  cattle  is  4  head  ?  14.  4.5  %  is  3f  %  ? 

5.  f  of  80  is  \  of  120?  15.  f  is  £  ? 

6.  380  pages  is  120  pages?  16.  1  is  .35? 

7.  $465  is  $130.20?  17.  8  is  .375? 

8.  $832  is  $807.04?  18.  1  T.  is  75  Ib.  ? 

9.  $1041.66f  is  $62.50?  19.  6  A.  is  5  sq.  rd.  ? 
10,  93  yd.  is  6.51  yd.?  20.  £11  is  £1  2s.? 


PERCENTAGE  213 

PROBLEMS. 

1.  A  farmer  raised  5390  bu.  of  grain  and  sold  1078  bu. 
What  per  cent,  of  it  did  he  sell  ? 

2.  A  merchant  having  375  yd.  of  cloth,  sold   150  yd. 
What  per  cent,  did  he  sell  ? 

3.  As  agent,  I  sold  a  house  for  $5000,  and  received  as 
remuneration  $50.     What  rate  per  cent,  did  I  receive  ? 

4.  A  lady  having  invested  funds  to  the  amount  of  $4750, 
on  withdrawing  the  money  received  $4987.50.     What  per 
cent,  did  she  gain  ? 

5.  If  8  Ib.  of  an  article  loses  4  oz.  in  weight  by  drying, 
find  what  per  cent,  of  water  escaped. 

6.  A  baseball  team  won   15   games   and  lost  9   games. 
What  per  cent,  of  its  games  did  it  win  ? 

7.  1-|-  times  a  number  is  what  per  cent  of  it  ? 

8.  If  $56.70  is  paid  for  the  use  of  $1260,  what  is  the 
rate  per  cent.  ? 

9.  A  boy  misspells  55  words  out  of  660.     What  per  cent, 
does  he  misspell  ? 

10.  .875  is  what  per  cent,  of  .125? 

11.  In  one  month  clover  seed  advances  from  $6.50  to  $7.00 
per  100  Ib.     What  was  the  rate  per  cent,  of  increase? 

12.  In  one  year  mixed  hay  advanced  from  $9  to  $11.75 
per  ton.     Find  the  rate  per  cent,  of  increase. 

13.  When  in  one  year  the  production  of  wool  in  the  United 
States  increased  from  298  million  Ib.  to  309  million  Ib.,  what 
was  the  rate  per  cent,  of  increase  ? 

14.  When  tallow  fell  from  5  cents  to  4-|-  cents  per  Ib.,  what 
was  the  rate  per  cent,  of  the  fall  ? 

15.  If  carpet  which  should  be  1  yd.  wide  is  only  34|-  in. 
wide.,  what  per  cent,  should  be  deducted  from  the  price  ? 

16.  What  per  cent,  of  1  rd.  3  yd.  2  ft.  5  in.  is  7  ft.  ? 


214  PRACTICAL   ARITHMETIC 

17.  What  per  cent,  of  MMMDLXX.  is  25  per  cent,  of 
MMDCCCLVI? 

18  Gas  is  reduced  from  $1.50  to  $1.00  per  M.  What  per 
cent,  of  the  original  cost  is  saved  ? 

19.  If  ^  of  a  ton  of  coal  is  sold  for  what  1000  Ib.  cost, 
what  is  the  gain  per  cent.  ? 

20.  The  cost  was  $3486,  the  selling  price  was  $4161.     Find 
the  gain  and  the  gain  per  cent.     Also  point  out  the  base,  the 
amount,  the  percentage,  and  the  difference. 

To  Find  the  Base. 

Since    Percentage    is    the    product    of    Base    and    Rate, 

obviously 

„         _  Percentage 
Rate 

EXERCISES. 
1.  $82  is  121  per  cent,  of  what  base? 

Process.  By  Analysis. 


100%  of  B.  =  M£M  =  $656. 

Explanation. 

The  percentage  is  $82.00  ;  the  rate  is  \1\%  ;  the  base  is  required. 

-p 
Since  the  base  is  required,  we  use  the  formula,  B.  =  —  -,  and  obtain 

B.  =  ---- 


2.  Find  of  what  number  : 

1.  385  is  12^%.  7.  168  men  is  8%. 

2.  396  is  11  %.  8.  462  oxen  is  7%. 

3.  250  is  15%.  9.  12  is  16 J %. 

4.  8.25  is  33^%.  10.  70  is  66f  %. 

5.  $64.36  is  10%.  11.  300  is  33^%. 

6.  38.6  bu.  is  13%.  12.  100  is  62|%. 


PERCENTAGE  215 

13.  72  is  44f%.  20.  5  cwt.  is  40%. 

14.  48  is  371%.  21.  75  yd.  is  18}%. 

15.  84  is  871%.  22.  $.50  is  31J%. 

16.  126  is  90%.  23.  160  is  106f  %. 

17.  24  is  81%.  24.  75  Ib.  is  61%. 

18.  10  is  61%.  25.  837  gal.  is  6%.      . 

19.  If  is  14%.  26.  9006  is  .06%. 

3.  $281.25  is  371%  of  what  number? 

4.  If  28%  of  a  number  =  $71.68,  what  is  the  number? 

5.  If  25%  of  a  number  =  $324,  what  is  40%  of  that 
number  ? 

PROBLEMS. 

1.  A  man  sold  a  horse  at  a  gain  of  $15,  which  was  15% 
of  the  cost.     Find  the  cost  and  selling  price. 

2.  A  farmer  sold  384  barrels  of  apples,  which  was  96% 
of  all  he  had.     How  many  had  he  ? 

3.  A  farm  was  sold  for  $536  less  than  cost,  which  was  at 
a  loss  of  20%.     What  was  the  cost  of  the  farm? 

4.  The  immigrants  of  a  population  number  143,000  per- 
sons, or  1 1  %  of  the  whole.     Find  the  total  population. 

5.  A  farmer  sold  110A.  43  sq.  rd.  of  land,  which  was 
20%  of  his  land.     How  much  land  had  he  at  first? 

6.  A  man  pays  $500  rent  a  year ;  85%  of  this  sum  is  33 J 
per  cent,  of  f  of  his  income.     Find  his  income. 

7.  On  the  sale  of  a  patent  $1600  was  lost.     What  was 
the  value,  if  the  rate  of  loss  was  16%  ? 

8.  A  sale  resulted  in  a  loss  of  $38.46,  which  was  J%  of 
the  cost.     Find  the  cost. 

9.  By  selling  an  article  for  66f  %  of  its  cost,  $23.25  was 
realized.     What  was  the  cost? 

10.  At  a  gain  of  -^%,  a  profit  of  $36  was  realized.     What 
was  the  cost? 


216  PRACTICAL  ARITHMETIC 

11.  30%  of  B.'s  money  is  in  a  bank,  and  50%  in  a  farm; 
the  remainder,  $2000,  is  in  P.  R.  R.  stock.  How  much 
money  does  B.  own  ? 

To  Find  the  Base  when  the  Rate  and  the  Amount  or 
Difference  are  Given. 

EXERCISES. 

1.  What  number  increased  by  6J%  of  itself  equals  510? 
Process.  Explanation. 

•  +         =  if  ,  the  number    +    V  of  the 


_  c-ir\        i    _  on    i  6  _  480  number  equals  ^|  of  the  number 

=  510;  and  since  j|  of  the  num- 
Or,  1   -f  .06-1-  --  LQ61  =  ber  =  510,  TV  of  the  number  = 

*in        ^10-1  OR9^  -  -  4SO  TT  of  51°  =  30  5  and  since  T6  = 

30,  H  =  16  times  30  or  480. 

Or,  since  once  the  number  and  .06^  times  the  number  =  510,  that 
number  must  equal  510  ^-  1.06J,  or  480. 

Hence  we  have  the  formula  : 

Amount  A. 

Base  =  l  or,  B.  =        --. 


2.  What  number  diminished  by  f  %  of  itself  equals  794  ? 

Process.  Explanation. 

|%  =  T§1F.  &$»  the   Dumber,  —  ^  of  the 

40.0  _     _3        _  .39.1  —  7Q4  num'  =  H£   of  the   num-  =  794> 

and  since  f^  of  the  num.  =  794, 

T0"0   -~2.  ^  __  ^  Of  794  —  2;    and  since 

4J<L  =  800.      Or,  ^  --=--  2,  ffi  =  400  times  2  or  800. 

^  _    0075  =    9925  ^r'    s^nce   °nce   the   num.  —  .0075 

7CM  QQ9^  __  «nO  times  the  num.  =  .9925  times  the 

num.,    and    since   .9925    times    the 
num.  =  794,  that  num.  must  equal  794  -4-  .9925,  or  800. 

Hence  we  have  the  formula  : 

Base  =  Difference          B  = 


1  —  Bate  1  —  E. 


PERCENTAGE 


217 

2%  of  itself  =516? 
16f%  of  itself  =  |l? 
16f%  of  itself  =1400? 
6J  %  of  itself  =  901? 
16%  of  itself  =261? 
3%ofitself=2f? 
140  #  of  itself  =630? 
1%  of  itself  ==  1897? 

12%  of  itself  =$616? 
14f%  of  itself  =96? 
36%  of  itself  =336? 

of  itself  =1? 

of  itself  =675? 


3.  What  number  increased  by  : 

1.  15%  of  itself  =4830?         9. 

2.  27%  of  itself  =508?         10. 

3.  33 J%  of  itself  =984?       11. 

4.  16f%  of  itself  =658?       12. 

5.  62J%  of  itself  =1820?      13. 

6.  10%  of  itself  =15,400?      14. 

7.  15%  of  itself  =690?          15. 

8.  8j%  of  itself  =4140.15?   16. 

4.  What  number  diminished  by  : 

1.  25%  of  itself  =1200?    '     6. 

2.  55%  of  itself  =1240?         7. 

3.  331%  of  itself  =  1260  ?       8. 

4.  11%  of  itself  =4539?        9. 

5.  3%  of  itself  =2667.50?    10. 


PROBLEMS. 

1.  A  man  owes  $14,300,  which  is  8%  more  than  his  prop- 
erty is  worth.     Find  the  value  of  his  property  ? 

2.  In  5  years  the  population  of  a  town  has  increased  25  % . 
The  population  is  now  7675.     WThat  was  it  5  yr.  ago  ? 

3.  A  regiment  lost  6J%  of  its  men,  and  then  had  left  750. 
Find  the  original  number. 

4.  Having  sold  30%  of  my  land,  I  had  then  34  acres 
remaining.     How  many  acres  had  I  at  first  ? 

5.  Two  farms  contain  630  acres,  and  one  farm  is  25% 
larger  than  the  other.     Find  the  size  of  each  farm  ? 

6.  What  was  the  cost  of  hats  sold  at  $1.00  apiece,  which 
was  at  a  loss  of  1 6-|  %  ? 

7.  A  horse  was  sold  for  $112,  and  the  loss  was  33J%. 
What  did  he  cost  ? 

8.  Silk  sold  at  $5.50  a  yd.  brings  in  a  profit  of 
What  did  it  cost? 


218  PRACTICAL  ARITHMETIC 

9.  A  wheat  speculator  lost  at  one  time  10%  of  his  money, 
and  at  another  time  12^%  of  the  remainder,  and  had  then 
$50,400  left.  How  much  had  he  previous  to  those  losses  ? 

1 0.  A  farmer  sold  two  cows  for  $50  apiece.     On  the  one  he 
gained  25%  ;  on  the  other  he  lost  25%.     Did  he  gain  or  lose 
by  the  sales,  and  how  much  ? 

11.  An  army  lost  25%  of  its  men  in  battle,  and  25  per 
cent,  of  the  remainder  were  discharged.     2223  men  still  re- 
mained.    Find  the  original  number  of  men  in  the  army. 

12.  A  farmer  having  sold   110  A.  43  sq.  rd.  of  land  had 
80%  left.     How  much  had  he  at  first? 

13.  A  patent  was  sold  for  $8000.     The  seller  lost  84%  of 
the  original  value.     Find  the  original  value  ? 

14.  A  flock  of  sheep  was  increased  by  250%  of  itself,  and 
then  numbered  1400.     Find  the  original  number. 

15.  How  many  Ib.  of  tallow  must  be  mixed  with  9^-  Ib. 
of  rosin  that  the  mixture  may  contain  24%  of  tallow? 

16.  By  selling  goods  at  60  cents  a  pound,  8%  is  lost.    What 
advance  must  be  made  in  the  price  to  gain  15%  ? 


REVIEW   EXERCISES. 

3.  B.  = 

R. 


1.  P.  =  B.  X  R.  2.  R.  =  ?:.  3.  B.  =  R 


4.  B.  =  i_^.     .-.  A.  ==  B.  X  (1  +  R.). 

1  -j~  -K». 

5.  B.  =      P     .     .-.  D.  -  B.  X  (1  —  R.)- 

1  —  R. 

1.  Show  how  formulae  2  and  3  are  derived  from  formula  1. 

2.  Show  how  formulae  4  and  5  are  derived. 

3.  Invent  five  problems  to  which  above  formulae  may  re- 
spectively apply. 


PERCENTAGE 


219 


Find  the  value  of  (?) : 


Kate  %  . 

Base. 

Percentage. 

Amount. 

Difference. 

1 
2 

16$ 
15 

$10.35 
2500  cd. 

9 
9 

9 

9 

3 
4 

12* 

831 

9 
9 

246  tons. 
7550  T 

9 

9 

5 

% 

365  da. 

29|da 

6 

7 

A 

9 

$16,200 
$10,800 

9 

$19,200 

*) 

.      .      . 

8 
9 
10 

9 
9 

871 

9 

5760  gr. 
196  Ib. 

144  cu.  ft. 

9 
9 

1728  cu.  ft. 
7000  gr. 

.      .      . 

11 
12 

9* 
23 

9 
9 

328JV 

9 

9 

9 

10,318 

13 

30 

9 

9 

$2924 

14 

15 
16 
17 

9 

9 

1 
37i 

$10,000 

9 

3 
2 

$120 

9 
9 
9 

•^ 

872 

9 

9 
li 

18 

2 
9 

1 

2 

4 

19 
20 
21 

9 
9 

9 

1  bu. 

9 

9 

5 

8  qt, 
24 
300 

144 

k  )     ' 
_1    Jt 

645 

REVIEW   PROBLEMS. 

1.  If  a  merchant  sells  f  of  an  article  for  what  J  of  it 
cost,  what  is  his  gain  per  cent.  ? 


Suggestion  :  Find  the  value  of  P.  and  use  Formula  2. 


220  PRACTICAL  ARITHMETIC 

2.  If  I  buy  a  farm  for  $5950,  for  what  must  I  sell  it  to 

gain  5%? 

Suggestion  :  Use  Formula  4. 

3.  I  received  for  my  horse  $164,  and  thereby  lost  18%  of 
his  value?     What  was  his  value? 

4.  If  I  buy  boots  at  $3.25  a  pair  and  sell  them  at  $3.87J 
a  pair,  what  per  cent,  do  I  gain  ? 

5.  In  a  school,  77  pupils  are  present,  which  is  87J%  of 
the  number  enrolled.     Find  the  number  enrolled. 

6.  f  is  1%  of  what  number? 

7.  A  man  sold  flour  at  $11   per  bbl.,  by  which  he  gained 
37|%  of  the  cost.     He  raised  the  price  to  $13.50.    What  does 
he  gain  per  cent,  by  this  advance? 

8.  If  the  bread  made  from  a  barrel  of  flour  weighs  33 \% 
more  than  the  flour,  what  is  the  weight  of  the  bread  ? 

9.  A  regiment  lost  in  a  campaign  400  men  out  of  965. 
What  was  the  rate  per  cent,  of  loss  ? 

10.  I  shall  be  obliged  to  use  to-day  53|%  of  my  money  to 
pay  a  note  for  $8620.     How  much  money  have  I  ? 

11.  In  a  certain  school,  during  1895,  the  attendance  of 
pupils  was  824,  which  was  3%   more  than  in  1894.     What 
was  the  attendance  in  1894? 

12.  In  1897  I  gained  in  business  $9207,  which  was  1% 
less   than    I    gained   in    1896.      How  much  did   I  gain  in 
1896? 

13.  In   a  mixture  of  copper  and   zinc,  the  copper  is   1^ 
times  the  zinc.     Find  the  percentage  of  each  ingredient  in  the 
mixture. 

14.  What   per  cent,  of  a  pound   troy  is  a  pound  avoir- 
dupois ? 

15.  A  piece  of  land  is  bought  for  $3600,  and  a  man  who 
owns  J-  of  it  sells  \  of  his  share  for  $800.     What  rate  per 
cent,  does  he  gain  ? 


COMMERCIAL  DISCOUNT  221 

COMMERCIAL   DISCOUNT. 

1.  Commercial  Discount  means  deduction  from  the  price 
of  merchandise  or  from  the  amount  of  a  bill,  and  is  computed 
at  some  rate  per  cent. 

2.  When  two  or  more  discounts  are  allowed,  the   first  is 
counted  off,  and  the  second  from  the  remainder,  and  so  on. 

3.  (1.)  The  Price  or  Bill"  is  the  Base.     (2.)  The  Kate  per 
cent,   is   the   Rate.     (3.)    The   Discount   is   the    Percentage. 
(4.)  The  Price  or  Bill  —  the  Discount  =  the  Net  Price  or 
Net  Amount  =  the  Difference. 

Hence  formulae  1  and  5  on  page  218  become  : 

1.  Discount  =  Price  or  Bill  X  Rate. 

2.  Net  Price  or  Net  Amount  =  Price  or  Bill  X  (1  —  BO- 

PROBLEMS. 

1%  What  is  the  net  amount  of  a  bill  of  $500  discounted  at 

20%? 

By  Formulae. 

Net  Amt.  =  Bill  X  (1  —  R.)  =  $500  X  (1  —  .20)  =  $500 
X  .80  =  $400. 

By  Analysis.  Brief  Process. 


20%  =  f 

£  of  $500  =  $100. 


500 
100 
400 


|500  —  $100  =  $400. 
2.  What  is  the  net  amount  of  a  bill  of  $500  with  33J 
and  121  Off  9 

Analysis.  Brief  Process. 

500.00 

Jof  $500  =  $166.66f; 
$500  —  $166.66f  =  $333.33J  ;       8 


J  of  $333.33J  =  $41.66$; 
$333.33J  __  $41.66|  =  $288.66f . 


333.33J 
41.66f 


288.66f 


222  PRACTICAL   ARITHMETIC 

3.  A  bicycle  marked  $90  was  sold  at  a  discount  of 
Find  the  discount  and  selling  price. 

4.  An  agent  sold  a  book,  price  $5.50,  at  20%  discount. 
What  did  he  get  for  it? 

5.  A  piano  was  marked  $650,  but  was  sold  at  a  discount 
of  20  and  8.     Find  the  selling  price. 

6.  Is  50  off  $2000  the  same  as  25  and  25  off?     Find  the 
difference,  if  any. 

7.  A  merchant  bought  $850  worth  of  goods  arid  received 
25  and  10  off.     What  was  the  net  amount  of  his  bill  ? 

8.  What  is  the  cash  value  of  a  bill  of  goods  amounting 
to  $2157.25  at  15  and  3  off? 

9.  Valentines  marked  $18  were  sold  at  20,  15,  and  -5  off. 
What  was  the  selling  price  ? 

10.  A  bill  of  goods  amounted  to  $3268.36.    What  was  the 
net  value  if  15  and  3  were  counted  off? 

11.  A  bill  of  envelopes  amounted  to  $86.40.     On  a  credit 
of  30  da.  the  reduction  was  ^  and  5  off;  for  cash   2J   off. 
With  all  rates  off,  what  was  paid? 

12.  Find  the  net  amount  of  a  bill  for  $762  subject  to  the 
following  discounts  :  40,  5,  and  10. 

13.  What  is  the  discount  on  a  bill  of  goods,  if  20%,  15%, 
and  5  %  are  successively  made  ? 

14.  Find  the  net  cost  of  20,000  bags  at  $4.50  per  M.,  with 
60,  10,  and  5  off. 

15.  Which  is  the  better  discount  for  the  buyer,  40  and  10 
off,  or  30  and  20  off  ? 

16.  After  getting  25%  off,  a  book  cost  me  $4.50.     What 
was  the  mark  price  of  the  book  ? 

17.  What  single  discount  is  equal  to  the  following  double 
discounts? 

1.  30  and  10%.  3.  50,  5,  and  5%. 

2.  40  and  15%.  4.  15,  10,  and  5%. 


GAIN  AND  LOSS  223 

18.  What  must  a  merchant  ask  for  an  article  which  cost 
$40,  so  that  he  may  deduct  20%  and  still  gain  10%  ? 

19.  A  bill  of  $670  is  subject  to  the  following  discounts: 
20,  15,  10,  5.     Find  the  net  amount  of  the  bill. 

20.  What  must  I  ask  for  an  article  that  cost  me  $3.60,  so 
that  I  may  deduct  12J%  and  still  make  a  profit  of  16f  %  ? 

GAIN   AND   LOSS. 

1.  Gain  or  Loss  in  business  is  computed  at  some  rate  per 
cent,  of  the  cost. 

2.  (1.)  The  Cost  is  the  Base.     (2.)  The  Rate  per  cent,  is 
the  Rate.     (3.)  The  Gain  or   Lpss  is  the  Percentage.     (4.) 
The  Selling  Price  (Cost  +  Gain)  is  the  Amount.     (5.)  The 
Selling  Price  (Cost  —  Loss)  is  the  Difference. 

3.  Hence  the  formula  on  page  218  become  : 

1.  G-.  or  L.  =  C.  X  B. 

2.  B.  =  O-  °r  L-.  4.  O.  =  -i^-  (gain). 

C.  1  -f-  B. 

o    ri         G-.  or  L.  ^    ~  S.  P. 

~—  5'  °'  = 


NOTE.  —  Let  the  pupil  derive  formulae  for  S.  P.  from  formulas  4  and  5. 

EXEBCISES. 

1  .  I  bought  corn  at  40  cents  a  bushel,  and  sold  it  at  a  gain 
of  12J%.     HOW  much  did  I  get  for  it? 

By  Formula.  By  Analysis. 

12J%  =  i    G.  =  C.  X  R.  12i%  =  i.     f,  cost,  +  i, 

G.  =  40  X  J  =  5.  gai'n?  —  f?  selling  price. 

40    +    5    =    45,    selling  f  ==  40  ;  |  =  5  ;  f  —  45, 

price.  selling  price. 

2.  I  bought  sugar  at  5  cts.  per  pound,  and  sold  it  at  5J  cts. 
per  pound.     ^Vhat  was  my  gain  per  cent.  ? 


224  PRACTICAL  ARITHMETIC 

By  Formula.  By  Analysis. 

5J  —  5  =  1,  gain.  5J  —  5  =  J,  gain.      Cost, 

R  =  GL  =  i  =   i  5,  ±=  100%  ;    1  =  20%. 

Gain>  *  = 


3.  I  sold  a  horse  and  lost  $50,  which  was  20%  of  the 
cost.     What  was  the  cost? 

By  Formula.  By  Analysis. 

Loss,  20%  -  $50.  1  %  = 
n  =  i,  Cost,  100%  = 
£f  4  =  $250. 

4.  I  sold  wheat  at  $1.00  per  bu.,  and  gained  12|%  of  the 
cost.     What  was  the  cost  per  bu.  ? 

By  Formula.  By  Analysis. 

Gain,  121%  =i.    f,cost,+ 

n    _    S-  P-    _     $1.00     __  i?  gain,  =  |^,  selling  price. 

1  4-  R.          1-1-  .12j  i  oo 

2.00  _  |-    $1.00;    i  =  ^J 


2.25  8.00          .QQ8 

f  =  -^-  *=  $.88|,  cost. 

5.  I  sold  hay  at  $10  per  ton  and  lost  10%  of  the  cost. 
What  was  the  cost  of  the  hay  ? 

By  Formula.  By  Analysis. 

s.  p.  10  Loss,  10%  =  TV     Cost, 

:    1— R.    "  "   1  — .10    " 
-£==$11.11+. 

-144  =  $11.11  +. 

6.  Cost,  $10,500;  rate,  20%.     Gain?     Selling  price ? 

7.  Cost,  $6  ;  rate,  8%.     Loss?     Selling  price? 

8.  Cost,  $8560;  selling  price,  $10,700.     Gain?     Rate? 

9.  Gain,  $6.30;  rate,  14%.     Cost?     Selling  price? 

10.  Cost,  $700;  rate,  15%.     Gain?     Selling  price? 

11.  Selling  price,  $19;  rate,  5%.     Loss?     Cost? 


GAIN  AND  LOSS  225 

12.  Selling  price,  $175;  rate,  30%.     Loss?     Cost? 

13.  Cost  12  cts.;  selling  price,  10  cts.     Loss?     Rate? 

14.  Cost,  9J  cts. ;  rate,  12}%.     Gain?     Selling  price? 

15.  Cost,  $.75;  selling  price,  $1.00.     Gain?     Rate? 

16.  Cost,  $1.00;  selling  price,  $1.25.     Gain?     Rate? 

17.  Cost,  $250;  rate,  35%.     Loss?     Selling  price? 
"18.  Cost,  $1.75;  selling  price,  $1.25.     Loss?     Rate? 

19.  Cost,  $.06  ;  selling  price,  $.05.     Loss?     Rate? 

20.  Gain,  $.12;  rate,  8%.     Cost?     Selling  price? 

21.  Gain,  10  cts. ;  rate,  10%.     Cost?     Selling  price? 

22.  Selling  price,  $180;  rate,  20%.     Cost?     Gain? 

23.  Selling  price,  $230;  rate,  8%.     Gain?     Cost? 

24.  Selling  price,  $4.56} ;  rate,  17%.     Loss?     Cost? 

25.  Selling  price,  $49.95;  loss,  $4.05.     Loss,  %?     Cost? 

26.  Gain,  $47.25;  rate,  7|%.     Selling  price ?     Cost? 

27.  Loss,  $38.46  ;  rate,  J%.     Cost?     Selling  price? 

28.  Cost,  $75.52  ;  rate,  3J%.     Gain?     Selling  price? 

29.  Selling  price,  $24.975  ;  loss,  $2.025.     Rate  ?     Cost  ? 

30.  Cost,  $1939.50;  rate,  |%.     Loss?     Selling  price? 

PROBLEMS. 

1.  A  man  sold  his  house  at  a  profit  of  15%.     If  he  paid 
$3000  for  it,  how  much  did  he  get  for  it? 

2.  What  per  cent,  is  lost  by  selling  tea  at  $.75  that  cost 
$1.00? 

3.  A  man  sold  a  horse  at  an  advance  of  $75,  which  was 
a  gain  of  25%.     What  was  the  cost  of  the  horse? 

4.  A  boot  and  shoe  dealer  lost  9%  by  selling  boots  at 
$3.75  a  pair.     What  was  the  cost  of  the  boots? 

5.  What  per  cent,  is  gained  on  goods  sold  at  double  their 
cost? 

6.  1  bought  a  horse  for  $500  and  sold  it  for  $300.    What 
per  cent,  did  I  lose  ? 


226  PRACTICAL  ARITHMETIC 

7.  The  selling  price  was  $30,  the  gain  25%.     What  was 
the  cost? 

8.  A  dry-goods  merchant  sells  goods  at  12  J  cts.  above 
their  cost  and  makes  a  gain  of  8%.     Find  the  cost. 

9.  By  selling  a  house  for  $3500  I  lose  $500.     What  is 
my  loss  per  cent.  ? 

10.  How  shall  I  mark  goods  that  cost  me  $1.00  a  yd.,  then 
deduct  1 5  %  from  that  mark,  and  still  realize  2  %  ? 

11.  I  bought  480  barrels  of  flour  at  $4.50  a  barrel  and 
sold  it  for  $2880.     Find  the  gain  per  cent. 

12.  A  cargo  of  flour  was  bought  for  $690.     For  what 
must  it  be  sold  to  gain  66f%  ? 

13.  I  sold  tea  for  114%  of  its  cost  and  made  a  profit  of 
$,07  a  Ib.     What  was  the  selling  price? 

14.  I  paid  $30  for  a  vase.     I  desire  to  gain  30%  on  it, 
after  dropping  40  %  from  the  asking  price.     What  price  shall 
I  ask? 

15.  When  4%  is  lost  on  cheese  sold  at  12  cts.  a  Ib.,  what 
was  the  cost  ? 

16.  I  sold  a  lot  of  goods  for  $200  and  thereby  gained 
15%.     Had  I  sold  them  for  $220,  what  per  cent,  would  I 
have  gained  ? 

17.  If  a  wagon  was  purchased  at  20%  less  than  $50,  and 
afterwards  sold  at  25%  more  than  cost,  at  what  price  was  it 
sold? 

18.  A  merchant  selling  goods  at  a  certain  price  loses  5%  ; 
but  if  he  had  sold  them  for  $20  more  he  would  have  gained 
3%.     What  did  the  goods  cost  him ? 

19.  If  a  merchant  sells  goods  for  f  of  their  cost,  what  per 
cent,  does  he  lose  ? 

20.  I  sold  a  quantity  of  potatoes  for  $850  which  cost  me 
$970.     What  per  cent,  did  I  lose? 

21.  An  agent  gets  a  discount  of  40%  from  the  retail  pnr?p 


COMMISSION  227 

of  articles  and  sells  them  at  the  retail  price.     What  is  his 
gain  per  cent.  ? 

22.  When  coal  was  sold  at  $4.56J  per  ton  there  was  a  loss 
of  17%.     What  was  the  cost? 

23.  A  druggist  gained  300%  by  retailing  quinine  at  $3.00 
per  ounce.     How  much  did  it  cost  him  per  ounce  ? 

24.  A  grain  dealer  sold  1380  bu.  of  wheat  at  $1.00  per  bu. 
and  lost  8%.     What  per  cent,  would  he  have  gained  had  he 
sold  at  $1.20  per  bu.  ? 

25.  A  drover  bought  100  cows  at  $20  a  head.     If  20  were 
killed  by  accident,  for  how  much  must  he  sell  the  remainder 
per  head  to  gain  25%  on  the  cost  of  the  whole  number  bought? 


COMMISSION. 

1.  Commission   is  the  percentage  allowed  an  agent  for 
buying  or  selling  goods  or  transacting  other  business. 

2.  Commission  is  computed  on  money  collected  by  him  or 
on  money  paid  out  by  him. 

3.  The  Collection  or  the  Payment  is  the  Base. 

4.  The  Rate  per  cent,  is  the  Rate. 

5.  The  Commission  is  the  Percentage. 

6.  The  Payment  -\-  the  Commission  is  the  Amount. 

7.  The  Collection  —  the  Commission  is  the  Difference. 
Hence  the  formulae  on  page  215  become  : 

1.  Com.  =  Coll.  or  Payt.  X  R. 


^ 
~  Coll.  or  Payt.' 

3.  Payt.  =  Amount 

1  +  R. 

4.  OOII     == 


1  —  R. 


228  PRACTICAL  ARITHMETIC 

PROBLEMS. 

1.  An  attorney  collects  a  debt  of  $500  on  a  commission 
of  3%.     What  is  his  commission? 

By  Formula.  By  Analysis. 

Com.  =  Coll.  X  R.  10?5== 

500  x  .03  =  $15.00. 

oyo  = 

2.  A  tax  collector  receives  $180  for  collecting  taxes  on  a 
3%  commission.     What  is  the  amount  collected? 

By  Formula.  By  Analysis. 

Com.  =  Coll.  X  R.       /.  Coll.  =  SS5t.  3%  =  $180. 


Coll.  =  :         =  =  $6000.  100%  =$6000. 

3.  Find  the  value  of  the  goods  that  can  be  purchased  for 
$420,  if  the  agent's  commission  is  5%. 

By  Formula.  By  Analysis. 

$420  =  the  Payment  +  the  Commis-         100%  =  Pay  t. 

sion  =  the  Amount.  5%  =  Com. 

Amt.              420  ,                105%  =  $420. 

Payt.  =  =  ppj  =     w  =  =  $400,  value  £  =  R 

of  goods.  100%  =  $400. 

4.  An  agent  sells  goods  at  21%  commission.     After  de- 
ducting his  commission,  he  remits  his  employer  $3763.50. 
How  much  money  did  he  collect  for  the  goods  sold  ? 

By  Formula.  By  Analysis. 

$3763.50   =   the   Collection  100%  =  Coll. 
—  the  Commission  —  the  2j-%  =  Com. 

Difference.  §1\%  =  $3763.50. 

Difference  3763.50  -  3763.50 


-  $3860.  100%  =.  -  $3860. 


COMMISSION  229 

5.  An  agent  sold  $2275  worth  of  goods  at  2%  commis- 
sion.    What  was  his  commission  ? 

6.  A  commission   merchant  received  $318.25  for  selling 
$12,730  worth  of  bankrupt  goods.     What  was  his  rate  of 
commission  ? 

7.  A  merchant   sent  his  agent  in  Cincinnati   $7000  to 
invest   in    pork,    after   deducting    his   commission   at   2J%. 
What  was  his  commission,  and  how  much  did  he  invest? 

8.  An  agent  received  a  certain  sum  of  money  to  invest  in 
goods  after  deducting  his  commission  of  3%.     He  invested 
$6250.     What  sum  did  he  receive? 

9.  If  I  send   my  agent  $4050  to  invest  in  goods  after 
deducting  3%  commission,  what  sum  will  he  invest? 

10.  What  is  the  commission  at  3%  for  selling  125  bbl.  of 
potatoes  at  $2.37J  per  bbl.  ? 

11.  A  commission  merchant  receives   2J  commission   for 
buying  grain  for  a  customer.     The  cost  of  the  grain  and  his 
commission  =  $4223.     How  much  does  the  grain  cost? 

12.  Find  the  amount  of  an  agent's  sales  when  his  commis- 
sion at  5%  =  $37.65. 

13.  A  real  estate  agent  collects  the  annual  rent  of  a  house 
and  retains  $13.25  as  his  commission  at  2J%.     What  is  the 
rental  of  the  house  ? 

14.  My  attorney  collected  a  bill  for  me  at  a  commission  of 
12J%  and  paid  me  a  net  sum  of  $56.     How  much  money 
did  he  collect? 

15.  An  agent  collected  20%  of  an  account  of  $860,  charg- 
ing 4%  commission.     Find  commission  and  sum  paid  over. 

16.  My  agent  collected  90%  of  a  debt  of  $5600  and  charged 
7J%  commission.     How  much  should  I  receive  from  him? 

17.  A  sale  of  real  estate  returned,  as  net  proceeds,  $2396.49, 
after  paying  $324.18  charges  and  a  commission  of  2%.     For 
how  much  did  it  sell  ? 


230  PRACTICAL  ARITHMETIC 

18.  Had  sold  for  me  500  bbl.  of  apples  at  $4.50  per  bbl., 
paying  2J%  commission ;  had  bought  for  me  with  the  pro- 
ceeds wheat  at  70  cts.  a  bu.,  paying  3%  commission.     How 
many  bu.  of  wheat  did  I  obtain  ? 

19.  How  much  commission  must  be  paid  to  a  collector  for 
collecting  an  account  of  $928.75  at  3f  %  ? 

20.  An  agent's  commission  for  selling  grain  was  $76.80,  at 
4%.     How  much  did  he  get  for  the  grain? 

21 .  A  real  estate  agent  sold  a  house  for  $7500  and  charged 
}%  commission.     Find  the  net  proceeds  of  the  sale. 

22.  An  agent  sold  goods  to  the  amount  of  $8725.     What 
was  his  commission  at  2J%  ? 

23.  A  consignee  sells  $6742  worth  of  woollen  goods,  charg- 
ing 2J%  commission  and  \\%  for  insuring  payment.     What 
sum  will  he  pay  over  to  the  consignor  ? 

24.  I  send  $10,000  to  my  correspondent  in  New  Orleans 
to   invest   in   cotton.      His   commission    is   \%  for  buying. 
What  sum  does  he  invest  and  what  is  his  commission? 

25.  A  man  receives  $1500  commission  on  his  yearly  sales. 
What  is  the  amount  of  his  sales  if  he  is  allowed  J%  commis- 
sion? 

26.  To  be  invested  in  cotton  at  15  cents  a  lb.,  $21,630.00; 
commission   allowed,    2J%  ;    marine   insurance   paid,    1J%  ; 
cartage  and  freight  paid,  1J%.     Find  the  sum  invested  in 
cotton  and  the  number  of  lb.  of  cotton  bought. 

REVIEW. 

Formulae  to  be  used. 

1.  I  sold  a  horse,  which  cost  me  $250,  at  a  loss  of  35%. 
What  did  I  get  for  him? 

2.  What  is  an  agent's  commission  on  the  purchase  of  an 
estate  for  $30,000,  at  \\%  ? 


COMMISSION  231 

3.  By  selling  a  watch  for  $19,  the  seller  loses  5%  on  his 
outlay.     What  would  have  been  his  loss  or  gain  per  cent,  if 
he  had  sold  the  watch  for  $23.75? 

4.  A  merchant's  prices  are  25%  above  cost  price;  if  he 
allows   a   customer    12%    on  his   bill,  what   profit   does   he 
make  ? 

5.  If  my  broker  buys  for  me  goods  worth  $13,000,  and 
his  commission  is  1J%,  how  much  must  I  pay  him? 

6.  A  speculator  sells  at  a  profit  of  75%,  but  his  purchasers 
fail,  and  only  pay  25  cents  on  a  dollar.     How  much  does  the 
speculator  gain  or  Jose  by  this  venture  ? 

7.  A  man  gains  15%  in  buying  an  article,  and  again  15% 
in  selling  it.     Find  the  whole  of  his  gain  per  cent. 

8.  If  goods  marked  at  45%  above  cost  are  sold  at  40% 
off,  what  is  the  gain  or  loss  per  cent.  ? 

9.  If  8%  be  lost  by  selling  an  article  for  $25.50,  what 
per  cent,  is  gained  or  lost  if  it  be  sold  at  $38.00  ? 

10.  A  carriage  is  sold  for  $175,  which  is  30%  less  than 
cost.     What  was  the  cost? 

11.  An  army  lost  18%  of  its  men  by  disease  and  desertion, 
and  then  lost  14%  of  the  remainder  in  battle;  the  number 
then  remaining  was  84,624.    Of  how  many  men  did  the  army 
consist  at  first  ? 

12.  If  I  sell  a  piano,  which  cost  $275,  for  $315,  what  is 
the  rate  per  cent,  of  gain  ? 

13.  If  I  buy  coffee  at  16  cents,  and  sell  it  at  20  cents  a 
pound,  what  per  cent,  do  I  make? 

14.  A  cargo  of  wheat  was  sold  for  $12,500,  by  which  a 
gain  of  25%  was  made.     What  was  the  amount  of  net  gain, 
after  paying  $150  freight  and  $75  for  other  charges? 

15.  If  the  commission  is  1 J  per  cent.,  what  bill  will  $3950 
buy? 

16.  An  auctioneer  sells  for  me  a  carriage  for  $140,  a  table 


232  PRACTICAL  ARITHMETIC 

for  $15,  50  yd.  of  carpet  at  60  cts.  a  yd.     His  commission  is 
2J%.     What  will  be  due  me  for  the  goods? 

17.  A  commission  merchant,  receiving  2J%  commission, 
had  410  bu.  of  potatoes  sent  him,  with  orders  to  sell  at  96 
cents  per  bu.     He  held  them  until  he  received  $492  above  his 
commission.     What  per  cent,  was  made  by  holding  them  ? 

18.  At  what  price  must  I  sell  goods  that  cost  $f  to  gain 
20%? 

19.  What  per  cent,  of  $90  is  33J%  of  $67.50? 

20.  Having  purchased  a  farm  for  $9000,  and  spent  $2500 
in  improvements,  I  sold  it  for  $13,800.     What  per  cent,  did 
I  make  on  my  investment? 

21.  A  man  sold  a  set  of  harness  for  $15,  and  lost  16f  %. 
If  he  had  sold  it  at  a  profit  of  20%,  what  would  he  have 
received  ? 

22.  W^hat  per  cent,  is  gained  by  selling  15  ounces  of  tea 
for  what  a  pound  costs  ? 

23.  A  speculator  sold  2760  bu.  of  wheat  at  $1.00  per  bu., 
and  lost  8  % .    How  much  per  cent,  would  he  have  gained  had 
he  sold  at  $1.20  abu.? 

24.  Which  is  the  better,  a  discount  of  25%  and  10%  off 
the  remainder,  or  a  discount  of  33 J%  off? 

25.  A  broker  sells  4000  bu.  of  wheat,  and,  after  deducting 
his  commission  of  2%,  remits  by  check   $4900.     At  what 
price  per  bu.  did  he  sell  the  wheat? 

26.  I  sent  a  commission  agent  500  bbl.  of  potatoes,  which 
he  sold  at  $2.50  per  bbl.     His  charges  were  :    commission, 
2J%  ;   storage,  1J%  ;   cartage,  $9.00.     How  much  was  due 
me? 

27.  A  drummer  earns  $3000  annually.     $1500  is  guaran- 
teed ;  the  remainder  is  his  commission,  at  5  % .     What  are  his 
annual  sales? 

28.  I  bought  1000  gross  of  screws  at  27  cents,  at  a  discount 


STOCKS  AND  BONDS  233 

of  15,  10,  and  5.  I  sold  the  lot  at  cost  plus  30%.  What 
was  my  gain  ? 

29.  Offered  cattle  for  sale  at  25%  above  cost,  but  was 
obliged  to  sell  them  for  14%  less  than  that  mark,  and  gained 
thereby  $170.  What  did  the  cattle  cost?  What  did  I  ask 
for  them?  How  much  did  I  sell  them  for? 

SO.  Received  $2020  to  buy  with ;  commission,  1  %  Find 
the  cost. 

31.  Collection,  $14,000  ;  commission,  $420.     Find  the  rate. 

STOCKS  AND  BONDS. 

DEFINITIONS. 

1.  Stock  is  invested  capital,  and  is  represented  by  certifi- 
cates which   attest  the  ownership   of  a  certain    number  of 
shares. 

2.  Bonds  are  written  obligations,  in  which  an  agreement  is 
made  to  pay  a  specified  amount  on  or  before  a  specified  date, 
with  interest. 

3.  The  Pace-Value  is  the  sum  mentioned  in  certificates  and 
bonds.     When  stocks  and  bonds  sell  for  their  face-value,  they 
are  said  to  be  at  par.     When  they  sell  for  more  than  their 
face- value,  they  are  said  to  be  at  a  premium.     When  they  sell 
below  their  face-value,  they  are  said  to  be  at  a  discount. 

4.  Coupons  certify  to  interest  due,  and  are  cut  off  and  sur- 
rendered when  the  interest  is  paid. 

5.  Bonds  are  issued  by  corporations  organized  under  law, 
and  take  their  name  from  the  name  of  the  corporation  that 

has  issued  them.     "  Buffalo  Railway — 4 — Q 84,"  means 

stock  issued  by  the  Buffalo  Railway  Company,  rate  4%,  in- 
terest payable  quarterly,  now  selling  at  $84  per  share, — i.e., 
at  a  discount. 

6.  Stock  Brokers  buy  and  sell  stocks  and  bonds ;  their 


234  PRACTICAL  ARITHMETIC 

commission  is  called  brokerage.     Brokerage  is  reckoned  at  \% 
or  \°/o  on  the  par  value. 

7.  A  Quotation  is  a  published  statement  of  the  current  sell- 
ing price  of  a  stock. 

Quotations. 
GOVERNMENT  BONDS.  STOCKS. 

1.  IT.  S.  4s,  registered,  1906,  110|.         6.  Adams  Express  4s,  105. 

2.  U.  S.  5s,  coupon,  1904,  U2|.  7.  Penna.  4  p.  c.,  110. 

3.  Currency  6s,  1899,  102].  8.  Schuylkili  E.  R.  5s,  105£. 

4.  Cherokee  4s,  1899,  101.  9.  N.  C.  Railway  4^s,  104i. 

5.  U.  S.  small  bonds,  105f .  10.   L.  V.  R.  R.  Coal  5s,  94. 

8.  The  Par  Value  is  the  Base. 

9.  The  Rate  of  Premium,  or  Discount,  is  the  Rate. 

10.  The  Premium,  or  the  Discount,  is  the  Percentage. 

11.  The  Quotation  Value  (ab.  par)  is  the  Amount. 

12.  The  Quotation  Value  (bel.  par)  is  the  Difference. 
Hence  the  formulae  on  page  218  become  : 

1.  Prem.  or  Disc.  =  P.  V.  X  R. 
Prem.  or  Disc. 


2.  R.  = 

3.  P.  V.  = 


P.  V. 

Prem.  or  Disc. 
R. 


Q.  V.  (ab.  par)        Q.  V.  (bel.  par) 
1  +  R.  1  —  R. 


Income  is  computed  on  the  face  of  a  bond,  at  the  face-rate.     5  shares 
($500)  U.  S.  4s  will  yield  $4  per  share,  or  $20.00  income. 

MODEL  SOLUTIONS. 

1.  Find  the  premium  on   25   shares  U.  S.  4s  quoted  at 
1  10  f  ;  also  the  annual  income  derivable  therefrom. 

U  Of  —  100  =  10f.     .'.  R.  =  10f^.     Par  value  of  25  shares  =  $2500. 
Formula:  Prem.  =  P.  V.  X  R-      $2500  X  -1075  ==  $268.75.     Income 
=  $2500  X  .04  =  $100. 


STOCKS  AND  BONDS  235 

2.  The  par  val.  of  U.  S.  bonds  =  $25,000,  and  the  premium 
=  §500.     Find  the  rate  and  the  quotation. 

Formula:  E.  =  ^-.  rf$fa  =  &  =  *%.  100#  +  2#  =  102 %, 
the  quotation. 

If  these  bonds  were  U.  S.  5s,  what  income  would  they 
yield  ? 

3.  I  bought  railroad  stock  quoted  at  96,  and  the  discount 
I  obtained  was  in  all  §24.     How  many  shares  of  stock  did  I 

buy? 

100  —  96  =  4.     .-.  K.  =  .04  =  4%. 

Formula :  P.  V.  =  5g£.     -^  =  $600  =  6  shares. 

If  this  is  6  %  stock,  what  is  my  yearly  income  therefrom  ? 

4.  I  bought  railway  stock  quoted  at  102^,  investing  $10,250. 
Find  the  number  of  shares  I  bought,  and  my  income  there- 
from at  5%. 

Formula :  P.  V.  =  £p£.     ^§  =  $10,000  =  100  shares. 
5%  of  $10,000  =  $500,  income. 

5.  If  you  buy  railway  stock  quoted  at  84,  and  invest  $3360, 
how  many  shares  will  you  buy,  and  what  will  be  your  income 
therefrom  if  the  stock  pays  4%  ? 

Formula :  P.  Y.  =  j%^-.      ^  =  $4000  =  40  shares. 

\Vhat  will  be  your  income  at  4%  ? 

6.  How  much  must  I  invest  in  6%  stock  at  102^-  to  secure 
me  an  annual  income  of  $300,  brokerage  J  ? 

Suggestions  :  "What  will  be  the  income  from  1  share  ?  How  many  shares 
will  yield  $300  ?  What  will  the  shares  cost  you  at  1021?  What  will  the 
brokerage  be  at  $£  per  share  ?  What  will  the  total  cost  of  the  investment 
be? 

7.  What  per  cent,   does  an  investment  in  Coney  Island 
Brooklyn  6s  offer  me  if  the  stock  is  quoted  at  140? 

Suggestions  :  $140  invested  yields  what  income?  That  income  is  what 
per  cent,  of  $140  ? 


236  PRACTICAL  ARITHMETIC 

8.  If  I  wish  to  obtain  7%  on  my  investment,  wnat  must  I 
pay  for  a  5  %  stock  ? 

Suggestion :  If  T^7  of  a  sum  =  $5.00,  what  must  that  sum  be?  Or, 
if  5  is  the  percentage  and  7  is  the  rate,  what  is  the  base  ? 

PROBLEMS. 

NOTE. — The  numbers  in  the  following  problems  refer  to  the  Bond  and 
Stock  Quotations  found  on  a  preceding  page.  Brokerage  is  not  considered 
unless  mentioned. 

1.  Find  the  market  value  of  150  shares  of  No.  1 ;  also 
the  income  and  the  rate  the  investment  pays. 

2.  Find  the  cost  of  investment  in  No.  10  to  secure  an 
income  of  $ 250. 

3.  Mr.  A.  owns  90  shares  of  No.  2.    What  is  his  income  ? 

4.  What  rate  do  Nos.  6,  7,  8,  9,  10  severally  pay  on  the 
investments  ? 

5.  Mr.  B.  has  $41,309  to  invest.     Which  will  secure  him 
the  larger  income,  No.  3  or  No.  4  ? 

6.  Which  will  give  him  the  higher  rate  on  his  investment? 

7.  How  many  shares  of  No.  3  can  he  buy,  paying  ^-% 

brokerage  ? 

Suggestion  :  Cost  of  a  share  =  102^  -(-  -J-. 

8.  If  I  wish  to  obtain  7%  on  my  investment,  what  must 
I  pay  for  a  6%  stock? 

9.  How  much  will  55  shares  C.  C.  C.  and  I.  R.  R.  stock 
cost  at  28},  brokerage  \%  ? 

10.  How  many  shares  of  railroad  stock  at  3%  discount  can 
be  bought  for  $2139.50,  brokerage  \%  ? 

11.  Which  is  more  profitable,  and  how  much,  to  invest 
$6000  in  6%  stock  purchased  at  75%,  or  5%  stock  purchased 
at  60%  ? 

12.  What  sum  must  I  invest  in  Louisiana  7s  at  107^  to 
secure  an  annual  income  of  $1750? 


STOCKS  AND  BONDS  237 

13.  Which  affords  the  greater  per  cent,  of  income,  bonds 
bought  at    125,  which   pay  8%,  or  bonds  which  pay  6%, 
bought  at  a  discount  of  10%  ? 

14.  At  what  price  must  I  purchase  15%  stock  that  it  may 
yield  the  same  rate  of  interest  as  6  %  stock  purchased  at  90  ? 

15.  What  is  the  cost  of  125  U.  S.  6s  at  104,  brokerage 

i%? 

16.  How  many  shares  must  a  broker  sell  to  realize  $10.50, 
commission  at  \%  ? 

17.  B  paid  $10,989  for  U.  S.  6s  at  110J,  brokerage  \%. 
What  was  his  income? 

18.  What  sum  must   be  invested  in  U.  S.   5s  at   116J, 
brokerage  J%,  to  secure  an  annual  income  of  $160? 

19.  What  is  the  annual  income  from  investing  $4446  in 
5J%  stock  at  92J,  brokerage  \%  ? 

20.  What  will  be  the  cost  of  17  shares  of  canal  stock  at 
93J  and  143  shares  gas  stock,  par  value  $10,  at  102f  ? 

21.  A  man  invested  $9562.50  in  the  stock  of  a  city  bank 
at  127J.     If  a  dividend  of  3J%  is  declared,  what  amount  of 
dividend  would  he  get? 

22.  6%  bonds  were  sold  at  118;  the  proceeds  were  in- 
vested in  4J%  bonds.     If  the  former  and  the  latter  incomes 
were   the   same,    at    what   quotation    were   the   4J%    bonds 
bought? 

23.  200  shares  of  stock,  par  value  $25,  and  sold  at  102J, 
J%  being  retained  for  brokerage,  how  much  is  paid  over? 

24.  Which  is  the  better  investment,  U.  S.  5s  at  98|%,  or 
U.  S.  6s  at  108f  %,  brokerage  \%  in  each  case? 

25.  Which  is  the  more  costly,  and   how  much  more,  15 
shares  of  N.  Y.,  N.  H.  &  H.,  at  85,  or  13  shares  N.  Y.  &  N.  E., 
at  102,  if  the  brokerage  in  each  case  is  \%  ? 

26.  How  much  money  must  be  invested  in  U.  S.  4Js  to 
yield  a  quarterly  income  of  $225,  bonds  selling  at  105 \  ? 


238  PRACTICAL  ARITHMETIC 

27.  A  man  invested  some  money  in  bonds,  at  par,  bearing 
6%  interest,  and  received  $300  semi-annually.     What  was 
the  sum  invested? 

28.  A  5%   stock   is  quoted   at   85  J.     A  purchaser   pays 
brokerage  at  \%.     What  rate  per  cent,  does  he  receive  on  his 
investment  ? 

29.  A  lady  would  secure  by  investment  an  annual  income 
of  $650.     How  much  5%  stock  must  she  buy  at  par  for  the 
purpose  ? 

30.  How  much  stock  can  be  bought  for  $14,178  when  the 
quoted  price  is  208 \  ? 

31.  Find  the  quoted  price  of  railroad  stock  when  the  cost 
of  250  shares,  including  brokerage  at  J%,  is  $30,312.50. 

32.  What  income  wall  $10,120  yield  if  invested  in  4% 
stock  bought  at  115? 

33.  If  a  6%  stock  is  quoted  at  120,  what  rate  per  cent, 
will  an  investor  receive  on  his  money  ? 

34.  If  I  invest  $1500  in  3%  stock  at  75,  what  is  my  in- 
come and  what  rate  per  cent,  do  I  get  on  my  investment  ? 

35.  If  I  exchange  48  shares  of  a  9%  stock  at   176   for 
U.  S.  4s  at  116J,  how  much  must  I  add  to  my  investment  to 
secure  the  same  income  ? 

36.  What  sum  of  money  must  be  invested  in  Louisville  & 
Nashville  Railroad  certified  gold  4%  bonds  at  84J  to  produce 
an  annual  income  of  $320,  brokerage  J  ? 

37.  If  $8000  5%  stocks  are  sold  at  90  and  the  proceeds 
invested  in  3J%  stocks  at  60,  find  the  increase  or  decrease  in 
income  ? 

38.  Find  the  price  of  a  3J%  bond  that  will  be  as  profitable 
an  investment  as  a  6  %  bond  at  par. 

39.  Mr.  A.  bought  U.  S.  6s  for  108,  kept  them  a  year,  and 
then  sold  them  at  118-|.     What  rate  of  interest  did  the  in- 
vestment pay  him  for  that  year  ? 


STOCKS  AND  BONDS  239 

40.  What  sum  must  be  invested  in  6%  stocks,  worth  95,  to 
yield  an  income  of  $4500  ? 

41.  Which  would  yield  the  larger  income,  §11,400  invested 
in  7%  stock,  at  95,  or  the  same  amount  invested  in  5%  stock, 
quoted  at  57  ? 

42.  At  \vhat  rate  must  a  5%  stock  be  sold  to  produce  8% 
on  the  investment? 

43.  If  I  buy  6%  stock  at  15%  discount,  what  is  the  rate 
of  interest  on  the  investment  ? 

44.  If  I  give  a  house  and  lot  worth  $2000  for  175  shares 
($10)  N.  Y.  Gas  Co.'s  stock,  what  is  the  rate  of  premium  ? 

45.  How  many  shares  of  bank  stock,  selling  at  5%  dis- 
count, can  be  bought  for  250  shares  of  insurance  stock,  selling 
at  14%  premium? 

46.  How  many  shares  of  stock,  par  25,  can  be  bought  for 
$2730,  when  quoted  at  105  ? 

47.  A  capitalist  bought  stock  at  65,  and  after  receiving  a 
dividend  of  5J%,  sold  it  at  82,  and  made  $1125.    How  much 
stock  had  he,  and  what  per  cent,  did  he  realize  ? 

48.  If  stock   is  bought  at  3J%   discount,  and  sold  at  a 
premium  of  2J%,  and  the  gain  is  $258.75,  what  is  the  par 
value  of  the  stock  ? 

49.  I  bought  bank  stock  at  96J,  and  sold  it  at  112J,  thereby 
gaining  $3556.     How  many  shares  were  there? 

50.  What  must  I  pay  for  6%  bonds  to  realize  5J%  on  my 
investment,  brokerage  \°/o  ? 

51.  In  order  to  realize  6%  annually  on  an  investment,  what 
must  I  give  for  bonds  that  pay  a  semi-annual  interest  of  3%, 
if  I  immediately  reinvest  the  semi-annual  interest  at  6%  ? 

52.  If  5%  bonds  are  bought  at  90,  what  is  the  rate  of 
income  on  the  investment? 

53.  A  lady  desiring  to  invest  money,  considered  5s  at  108, 
6s  at  124,  and  7s  at  129,     Which  was  preferable? 


240  PRACTICAL  ARITHMETIC 

54.  A  broker  charges  $25  at  \%  for  buying  Pennsylvania 
R.  R.  ($50).     How  many  shares  did  he  buy  ? 

INSURANCE. 

1.  Insurance  is  security  guaranteed  for  loss  by  fire  or  other 
specified  causes. 

2.  Property  Insurance  includes  : 

1.  Fire  Insurance.  ^    ^. 

n    ,  ,    .      T  Jrremmm    computed    as 

Z.  Marine  insurance.          Y 

,    T  .      0      .    T  percentage. 

3.  Live  Stock  Insurance.  J 

3.  Personal  Insurance  includes  : 

1.  Life  Insurance.          ^    ^  ,    , 

.  ,       T  Jrremium    computed    at    a 

2.  Accident  Insurance.  V  .  AI™™ 

certain  sum  per  $1000. 

3.  Health  Insurance.     J 

4.  The  written  agreement  is  called  the  Policy  ;  the  sum 
named  in  the  policy  is  called  the  Face  ;  the  sum  paid  annually, 
semi-annually,  or  quarterly  is  called  the  Premium. 

1.  The  Face  (the  amount  insured)  is  the  Base. 

2.  The  Rate  of  Premium  is  the  Rate. 

3.  The  Premium  is  the  Percentage. 
Hence  we  have  the  following  formulae  : 

1.  Pace  X  Bate  =  Premium. 

2.  Premium  -*-  Face  =  Bate. 

3.  Premium  -r-  Bate  —  Face. 

MODEL  SOLUTIONS. 

1,  How  much  will  it  cost  to  insure  a  house  worth  $3000 
at 


Formula:  Premium  =  Face  X  Rate  =  $3000  X  -fa  •—  137.50. 
2.  A  merchant  insures  his  store,  valued  at  $4850,  for  |-  of 
its  value  at  %%.     What  is  the  premium? 

|  of  4850  =  $3880.     |#  =  Tfo  =  .00875. 
Formula:  Face  X  Rate  =  $3880  x  ,00875  =  $33.95. 


INSURANCE  241 

3.  The  insurance  on  a  barn  at  -f-%  costs  $18.     What  is  the 
face  of  the  policy  ? 


Formula:  Face  =  Premium  -=-  Kate  =  18  -=-  .0075  =  $2400. 
Or,  \%  =  $18.     \%  =  $6.     \%  =  $24.     100^  =  $2400. 

If  I  pay  $30  insurance  on  a  $3000  house,  what  is  the  rate  ? 

Formula  :  Rate  =  ^ 


Or,  $3000  =  100#.     $1  =  -tffoft  =  &%.     $30  =  f$#  =1%. 

PROBLEMS. 

1.  If  a  man  pays  $30  insurance  at  1J%,  what  amount  of 
insurance  does  he  get  ? 

2.  A  vessel  and  cargo   valued  at  $2840  are  insured  at 
3J%.     What  is  the  premium? 

3.  A  man  has  a  house  worth  $5600.     He  insures  it  at 
\\%  on  fy  of  its  value.     Find  the  cost  of  insurance. 

4.  What  is  the  total  premium  on  a  house  worth  $4500 
insured  for  5  years  at  1  J  %  ? 

5.  How  much  is  the  premium  for  insuring  a  stock  of 
goods  for  $15,000  at  1J%? 

6.  Mr.  Jacobs  paid  $652.50  for  insuring  property  valued 
at  $43,500.     What  was  the  rate  ? 

7.  A  vessel  and  cargo  were  insured  for  f  of  their  value  at 
1J%.     The  premium  was  $2475.     At  what  price  were  the 
vessel  and  cargo  valued? 

8.  $3.75  was  the  premium  on  f  the  value  of  some  furni- 
ture at  1  %  a  year.     What  was  its  insurance  valuation  ? 

9.  One  company  offers  to  take  a  $12,000  risk  at  \\%  for 
five  years,  and  another  at  J%  a  year.     Which  is  the  cheaper? 

10.  An  insurance  company  loses  $3528  by  the  wreck  of  a 
carload  of  flour  which  it  had  insured  for  $3600.     What  was 
the  rate  of  insurance  ? 

11.  A  merchant  imports  a  cargo  from  Liverpool,  England, 

16 


242  PRACTICAL  ARITHMETIC 

worth  £1500  and  insures  it  at  -£%.     Find  the  premium  in 
U.  S.  money. 

12.  For  what  sum  must  a  policy  be  made  out  to  cover  the 
insurance  on  a  property  of  $2100  at  -f-%  ? 

13.  If  it  cost  $93.50  to  insure  a  store  for  oue-half  of  its 
value,  at  lf%,  what  is  the  store  worth? 

14.  A  person  insured  his  house  for  j  of  its  value  at  40 
cents  per  $100,  paying  a  premium  of  $73.50.     What  was  the 
value  of  the  house? 

15.  At  -£%,  how  much   insurance   can  be  effected   upon 
a  store  for  $108? 

16.  For  what  sum  should  a  cargo  worth  $74,496  be  insured 
at  3%  so  that,  in  case  of  loss,  the  owner  may  recover  both  the 
value  of  the  cargo  and  the  premium  paid  ? 

17.  A  man  has  a  house  worth  $5600.     He  insures  it  at 
1^%  on  %-  of  its  value.     Find  the  cost  of  insurance. 

18.  If  a  tax  of  $12  is  paid  on  a  house  and  lot  valued  at 
$1200,  what  is  the  rate  per  cent,  of  tax? 

19.  A  vessel  worth  $28,000  was  insured  at  If  %,  and  the 
cargo,  worth  $15,000,  at  2^%.    Both  were  totally  lost.    What 
was  the  loss  to  the  insurer? 

20.  A  man  25  years  of  age  has  his  life  insured  for  $6000 
at  $19.85  on  $1000  annually.     What  annual  premium  does 
he  pay  ? 

21.  If  a  man  35  years  of  age  takes  out  a  life  policy  for 
$8500  at  $22.70  on  $1000  annually,  and  dies  at  the  age  of  60, 
how  much  does  the  amount  insured  exceed  the  sum  of  the 
premiums  ? 

22.  If  Mr.  B.  takes  out  a  life  policy  for  $8000,  what  is  his 
yearly  premium  at  the  rate  of  $26.50  on  $1000? 

23.  At  the  age  of  28  years  I  took  out  an  endowment  policy 
for  $10,000.     What  is  my  yearly  premium  at  the  rate  of 
$45.15  on  $1000? 


DIRECT  TAXES  243 

24.  I  insure  my  life  for  $8000,  paying  $19.80  per  $1000 
per  year.     What  do  I  pay  the  company  if  I  live  20  years 
after  insurance? 

25.  If  a  person  who  is  insured  for  $5000,  at  an  annual 
premium  of  $28.90  per  $1000,  dies  after  9  payments,  how 
much  more  will  his  heirs  get  than  has  been  paid  in  premiums? 

26.  A  lady  insures  her  life  for  $8000,  at  an  annual  pay- 
ment of  $29.30  per  $1000.     If  she   lives   15  years,  what 
amount  will  she  have  paid  in  premiums? 

DIRECT  TAXES. 

1.  A  Tax  is  a  sum  of  money  levied  on  persons  in  behalf 
of  the  public  welfare. 

2.  A  Poll  Tax  is  levied  on  the  person.     A  Property  Tax 
is  levied  on  property. 

3.  Assessors  determine  the  value  of  property. 

4.  A  Tax-Collector  collects  the  taxes ;  his  salary  is  com- 
monly a  percentage  of  the  sum  collected. 

5.  Property  Tax  is  reckoned  at  some  rate  per  cent,  on  the 
value  of  the  property  assessed. 

MODEL  SOLUTION. 

A  tax  of  $15,600  is  to  be  raised  in  a  town  in  which  the 
taxable  property  is  $3,200,000 ;  there  are  1000  persons  who 
pay  a  poll-tax  of  $2.00  each.  What  is  the  rate  of  taxation  ? 
What  is  A.'s  tax,  whose  property  is  valued  at  $6000,  and 
who  pays  a  single  poll-tax? 

1.  The  poll-tax  =  $2.00  X  1000  =  $2000. 

2.  Total  tax,  $15,600  —  $2000  =  $13,600,  tax  to  be  raised  on  property. 

3.  $13,600  ~-  3,200,000  ==  .004£.     Kate  =  4£  mills  on  a  dollar. 

4.  A.'s  tax  =  $6000  X  -004^  =  $25.50,  on  property. 

5.  $25.50  +  $2.00  =  $27.50,  A.'s  entire  tax. 


244  PKACTICAL   ARITHMETIC 

Hence  the  formulae : 

1.  Bate  of  Taxation  =  [Total  Tax  —  Poll  Tax]  -=-  Total 
Valuation. 

2.  Each  Citizen's  Tax  =  His  Valuation  x  Rate  +  His  Poll 
Tax. 

PROBLEMS. 

1.  A  certain  town  wishes  to  raise   $1644  by  taxation. 
The  property  of  the  town  is  assessed  at  $224,000.     There  are 
400  polls,  assessed  at  $0.75  each.     What  is  the  tax  on  $1  ? 

2.  At  the  above  rate,  what  would  be  A.'s  tax  if  he  pays 
for  real  estate  valued  at  $3655,  for  personal  property  valued 
at  $980,  and  for  2  polls? 

3.  If  a  tax-collector  receives  $54  for  collecting  $1800, 
what  is  his  rate  of  commission? 

4.  A  tax-collector  receives  $180  for  collecting  taxes  on  a 
3%  commission.     What  is  the  amount  collected? 

5.  How    many   dollars    on    $1000    must    be    levied    on 
$597,600  to  raise  $5976  tax? 

6.  How  much  is  a  man  taxed  who  was  assessed  for  one 
poll    $0.75,    and   on    property   valued    at    $5390,   the    rate 
being  \%  ? 

7.  In    a    town    whose    taxable    property    is    valued    at 
$5,463,000  a  tax  of  $9560.25  is  raised.     What  is  the  rate 
of  taxation? 

8.  My  property,  which  cost  me  $7800,  is  taxed  at  f  of  its 
value.     If  my  tax  is  $15.60,  what  is  the  rate  of  taxation? 

9.  What  sum  must  be  assessed  to  raise  $3750,  besides 
paying  2%  for  collection  ?    What  would  be  the  taxable  valua- 
tion of  property  to  raise  that  sum  if  the  rate  were  .003275  ? 

10.  A  tax  of  $14,250  is  to  be  assessed  on  a  town ;  the  real 
estate  is  valued  at  $1,200,000  and  the  personal  property  at 
$750,000 ;  there  are  400  polls,  each  of  which  is  taxed  $1.50. 
What  is  the  rate  of  taxation  ? 


DIRECT  TAXES 


245 


11.  What  is  the  assessed  value  of  property  taxed  $87.50 
at  the  rate  of  5  mills  on  a  dollar  ? 

12.  A  tax  of  $28.50  is  to  be  raised  on  a  town,  and  suffi- 
cient besides  to  pay  for  collecting  at  5%.    If  the  rate  is  ^  cent 
on  a  dollar,  what  is  the  property  worth  ? 

13.  In  a  certain  district  a  school-house  is  to  be  built  at  a 
cost  of  $18,527.     What  amount  must  be  assessed  to  cover 
this  and  the  collector's  fees  at  3%  ? 

14.  Find  the  entire  tax  that  must  be  assessed  in  order  that 
a  town  may  receive  $12,134  after  the  collector  deducts  his 
commission  of  2^%. 

After  the  tax  rate  has  been  determined,  the  computation  of 
a  tax  list  is  facilitated  by  the  use  of  a  table. 

Table. 
Rate,  $0.015. 


PROP. 

TAX. 

PROP. 

TAX. 

PROP. 

TAX. 

$1    ... 

.015 

$4  ... 

.06 

&7 
f  I    .  .   . 

.105 

2  ... 

.03 

5  ... 

.075 

8  ... 

.12 

3  ... 

.045 

6  ... 

.09 

9  ... 

.135 

15.  By  using  the  table,  find  the  tax  on  $8450. 

Process. 

Tax  on  $8000  =  $120.00,  1000  times  .12. 
Tax  on      400  =        6.00,  100  times  .06. 
Tax  on        50  =          .75,  10  times  .075. 
$126.75. 

16.  In  like  manner  find  the  tax  of: 

1.  C.  H.  Anheier,  on  $910. 

2.  R.  B.  Bates,  on  $2356. 

3.  G.  B.  Caldwell,  on  $3600. 


246  PRACTICAL    ARITHMETIC 

4.  M.  F.  Dooley,  on  $9855. 

5.  Z.  S.  Eldridge,  on  $10,864,  paying  2  polls,  at 

$1.50. 

6.  J.  S.  Escott,  on  $20,200,  paying  1  poll,  at  $1.50. 

7.  S.  R.  Flynn,  on  $31,750,  paying  3  polls,  at  $1.50. 

8.  E.  J.  Graham,  on  $111,368,  paying  2  polls,  at 

$1.50. 

9.  C.  P.  Hatch,  on  $200,500,  paying  5  polls,  at  $1.50. 
10.  E.  J.  Johnson,  on  $567,005,  paying  2  polls,  at 

$1.50. 


INDIRECT  TAXES. 

1.  Indirect  Taxes  are  levied  upon  merchandise.     They 
consist  of  Duties,  levied  on  imported  goods,  and  of  Internal 
Revenue,  levied  on  domestic  goods. 

2.  Duties  are  of  two  classes, — Specific  and  Ad  Valorem. 

3.  Specific  Duties  are  levied  on  each  yard,  pound,  etc., 
of  the  article.     Ad  Valorem  Duties  are  levied  at  a  rate  per 
cent,  of  the  cost  of  the  article  in  the  country  in  which  it  was 
bought. 

4.  Specific  Duties  are  computed  on  the  net  measure  or  weight, 
Tare  being  allowed  for  the  weight  of  box  or  wrappings,  and 
for  Breakage,  Leakage,  etc. 

PROBLEMS. 

NOTE — In  the  examples  that  follow,  the  present  tariff  rates  (1899)  are 
used. 

1.  What  is  the  duty  on   1250  Ibs.  of  desiccated  apples 
imported  at  the  rate  of  2  cents  per  pound  ? 

Process. 
1250  X  -02  ==  $25.00,  duty. 


INDIKECT  TAXES  247 

2.  A    merchant   imported    1000    yd.    of  Brussels   carpet 
costing  in  Europe  3  shillings  per  yd.     What  was  the  duty 
at  40%  ? 

Process. 

£1  db  $4.8665;  Is.  =  $.243325;  3s.  =  .729975.    .-.  The 
cost  of  1000  yd.  =  1000  X  .729975  =  $729.975. 
40%  of  $729.975  =  $291.99,  duty. 

3.  A   merchant   imported   $1250  worth   of  silk   beaded 
goods.     What  was  the  duty  at  60%  ? 

4.  What  is  the  duty,  at  3  cts.  per  lb.,  on  175  bags  of 
coffee,  each  containing  115  lb.,  valued  at  20  cts.  per  Ib. 

5.  Find  the  duty  on  100  boxes  of  Castile  soap,  containing 
each  110  lb.,  costing  20  liras  per  cwt.,  at  1^  cts.  per  lb.,  tare 
allowed,  5%. 

6.  What  is  the  duty  on  400  boxes  of  cigars,  each  box 
containing  500  cigars,  gross  weight  400  lb.,  costing  80  cts.  per 
lb.   in  Havana,  at  the  rate  of  $4.50  per  lb.  and  25%   ad 
valorem,  together  with  the  internal  revenue  tax  of  $3.00  per 
1000  cigars? 

7.  If  an  imported  piano  cost  in  Europe  $200  and  was 
subject  to  a  duty  in  New  York  of  45%,  at  what  price  must  it 
be  sold  to  gain  25%? 

8.  What  is  the  duty,  at  21  cts.  a  pound,  on  3750  lb.  of 
coffee,  allowing  5%  for  tare? 

9.  What  is  the  duty  on  500  lb.  of  raisins,  in  boxes,  valued 
at  10  cts.  a  pound,  allowing  15%  for  tare,  when  the  duty  is 
2^-  cts.  a  pound  ? 

10.  What  is  the  duty,  at  If  cts.  per  pound,  on  7  T.  of 
steel  anvils,  of  2240  lb.  each,  invoiced  at  20  cts.  a  pound  ? 

11.  An  importer  paid  duties  amounting  to  $386.75.     If 
the  duty  was  25%  of  the  value  of  the  goods,  what  was  their 
value  ? 


248  PRACTICAL  ARITHMETIC 

12.  What  will  be  the  duty,  at  55  cents  per  sq.  yd.,  on  6 
pieces  of  cloth,  each  containing  54  yd.,  32  in.  wide? 

13.  A  merchant  imported  from  Havana  25  hhd.  of  W.  I. 
molasses,  which  was  invoiced  at  40  cents  per  gal.     Allowing 
J  %  for  leakage,  what  was  the  duty  at  6  cents  a  gallon  ? 

14.  What  is  the  duty,  at  44  cents  per  lb.,  and  55%  ad 
valorem,  on  700  yd.  of  cloth,  invoiced  at  $1.60  per  yd.,  one 
yd.  weighing  1^  lb.  ? 

15.  The  duty  on  certain  cotton  goods  is  5J  cents  per  sq. 
yd.,  and  20%  ad  valorem.     Find  the  duty  on  267  pieces,  30 
in.  wide,  each  piece  containing  37  yd.,  and  costing  7  cents 
per  yd. 

16.  I  imported  100  tons  of  iron,  costing  IJd.  per  lb.,  on 
which  I  paid  a  duty  of  $4.00  per  ton.     The  freight  was  6s. 
per  ton.     What  was  the  entire  cost  in  U.  S.  currency  ? 

17.  An  importer  bought  1000  pieces  of  certain  goods  at 
$40  per  piece ;  the  duty  thereon  was  50%  ;  the  freight,  etc., 
was  $1200.     How  must  the  goods  be  sold  to  gain  25%  ?" 

18.  A  quantity  of  bookbinders'  calf-skins  cost  $630,  in- 
cluding $15  for  freight  and  $102.50  for  duty.    What  was  the 
rate  ad  valorem  ? 

19.  If  the  importation  of  83  J  doz.  of  gloves  doubled  their 
cost,  which  was  50  fr.  per  dozen,  what  was  gained  on  each 
pair,  and  on  the  entire  lot  by  selling  them  at  $2.00  per  pair  ? 

20.  Find  the  total  cost  of  glassware  on  which  $311.85  for 
duty  at  45%  ad  valorem  was  paid,  and  16%  for  breakage  was 
allowed. 

21.  If  the  gross  cost  is  $2630,  the  freight  $100,  the  duty 
$330,  what  is  the  rate? 

22.  100   pieces  of  French   goods   were   invoiced   at   $40 
per  piece ;  the  duty  paid  was  50%  ;  the  freight,  etc.,  amounted 
to  $1500.     How  must  they  be  sold  to  gain  20%  ? 

23.  The  invoice  price  of  goods  is  $1.00  per  yard;  the  ad 


INTEREST  249 

valorem  duty  is  20%  ;  the  specific  duty  is  $0.20.     Find  the 
gross  value  of  a  single  yard. 

24.  The  specific  duty  is  $0.44  per  Ib. ;  the  ad  valorem 
duty  is  60%  ;  the  gross  cost  is  $244.     Find  the  invoice  price 
of  100  Ib. 

25.  The  ad  valorem  duty  is  50%  ;   the  invoice  price  is 
$500;  the  selling  price  at  a  profit  of  25%  is  $1000  on  100 
Ib.     Find  the  gross  value  and  the  specific  duty. 


INTEREST. 

1.  Interest  is  money  paid  for  the  use  of  money,  and  de- 
pends both  upon  a  certain  rate  per  cent,  and  the  length  of 
time  the  money  is  in  use. 

2.  The  money  used  is  the  Principal  (Base). 

3.  The  interest  for  one  year  is  the  Percentage. 

4.  The  interest  for  a  longer  or  a  shorter  time  than  one  year 
is  the  product  of  the  percentage  and  the  time  expressed  in 
years  or  in  the  fraction  of  a  year. 

5.  The  time,  when  expressed  in  months,  must  be  divided 
by  12 ;  when  expressed  in  days,  by  360. 

6.  Since  percentage  equals  the  product  of  principal  (base) 
and  rate,  we  have  the  following  formulae : 

1.  Interest  =  Principal  X  Bate  X  Years ; 
or,  (Int.  =  Pr.  X  B.  X  Y.). 

2    Interest Principal  X  Rate  X  Months . 

12 


or,  (int.  = 


Pr.  X  R.  X  mo/ 
12 


3    intercot  =  Principal  x  Rate  X 

36O 


250  PRACTICAL   ARITHMETIC 

MODEL  SOLUTIONS. 

1.  What  is  the  interest  of  $550  at  5%  for  4  yr.? 

Process. 
Int.  =  Pr.  X  K.  X  Y.  =  $550  X  .05  X  4  =  $110. 

Explanation. 

Since  the  int.  is  required  for  exactly  4  yr.,  we  use  the  formula,  Int.  = 
Pr.  X  B.  X  Y.  =  $550  X  .05  X  4  =  $110. 

Or,  we  may  explain  thus : 

Since  the  rate  per  cent,  is  5,  the  int.  for  1  yr.  =  .05  of  $550  =  $27.50. 
Since  the  int.  for  1  yr.  =  $27.50,  the  int.  for  4  yr.  =  4  times  $27.50  = 
$110. 

2.  What  is  the  interest  of  $500  for  5  yr.  and  2  mo.  at  6%  ? 

Process. 

5  yr.  2  mo.  =  62  mo. 
31 

Int.  =  — : ^ — —  —  =  — ^ =  5  X  31  =  $155. 

**  W 

m 

Explanation. 

Since  the  int.  is  required  for  62  mo.,  we  use  the  formula,  Int.  = 
Pr-x^2xmo-  =  5QOX1fx62.  By  cancellation  we  have  5  X  81  =  $155. 

Or, 

The  int.  of  $500  for  1  yr.  at  \%  =  $5.00;  at  6#,  therefore,  it  =  $5  X 

31 

6,  for  1  yr. ;  for  1  mo.  it  =  ^-5-*-6  ;  for  62  mo.  it  =  *5  X  f^X  ^  =  $155. 
1 2 


3.  What  is  the  interest  of  $222.50  for  10  yr.  8  mo.  21  da, 
at  3%  ? 


INTEREST  251 

Process. 
10  yr.    =  120  mo.  ^ 

8  mo.^        8  mo.  U  128.7  mo.    Int  =  Fr"  X  ^  X  m°'. 
21  da.    =  0.7  mo.  j 

.55625 

=  mm*J»  x  128-7  =  .55625  X  128.7  =  $71.59. 

m 

Or, 

10  yr.    =3600  da.  -) 

8  mo.  =  240  da.  V  =  3861  da.    Int.  =  -    X360X 
21  da.   =     21  da.  j 
22250  X__XJ8«1  .89 


NOTE.  —  The  work  may  be  simplified  by  removing  decimals  before  can 
celling. 

EXERCISES. 
Int  =  Pr.  X  B.  X  Y. 

1  .  Find  the  interest  of  : 

1.  $100  for  1  yr.  at  8%.  6.  $600  for  6  yr.  at  10%. 

2.  $200  for  2  yr.  at  6%.  7.  $700  for  7  yr.  at  7%. 

3.  $300  for  3  yr.  at  10%.  8.  $800  for  8  yr.  at  6%. 

4.  $400  for  4  yr.  at  7%.  9.  $900  for  9  yr.  at  6%. 

5.  $500  for  5  yr.  at  6%.  10.  $1000  for  10  yr.  at  8%. 

TV*     -  Pr-  X  R.  X  mo. 
^2~ 

2.  Find  the  interest  of  : 

1.  $590  for  3  yr.  7  mo.  at  7%. 

2.  $600  for  4  yr.  11  mo.  at  10%. 

3.  $830  for  5  yr.  10  mo.  at  6%. 

4.  $950  for  5  yr.  9  mo.  at  6%. 

5.  $1070  for  6  yr.  11  mo.  at  6%. 

6.  $470  for  3  yr.  8  mo.  at  10%. 

7.  $2359  for  4  yr.  7  mo.  at  12%. 


252  PRACTICAL   ARITHMETIC 

8.  $3597  for  6  yr.  9  mo.  at  8%. 

9.  $2300  for  4  yr.  6  mo.  at  6%. 
10.  $7000  for  1  yr.  7  mo.  at  7%. 

3.  Find  the  interest  of : 

1.  $10  for  1  yr.  1  mo.  3  da  at  5%. 

2.  $121  for  2  yr.  2  mo.  6  da.  at  6%. 

3.  $25.16  for  3  yr.  3  mo.  9  da.  at  6%. 

4.  $36.24  for  4  yr.  4  mo.  12  da.  at  7%. 

5.  $48.20  for  5  yr.  5  mo.  15  da.  at  7%. 

6.  $2000  for  6  yr.  6  mo.  18  da.  at  6%. 

7.  $590.50  for  7  yr.  7  mo.  21  da.  at  6%. 

8.  $640.82  for  8  yr.  8  mo.  24  da.  at  .10%. 

9.  $725.83  for  9  yr.  9  mo.  27  da.  at  6%. 
10.  $618.24  for  10  yr.  10  mo.  3  da.  at  6%. 

T   ,    ._  Pr.  X  R.  X  da. 
360 

4.  Find  the  interest  of: 

1.  $7000  for  1  yr.  6  mo.  7  da.  at  3%. 

2.  $8300  for  2  yr.  5  mo.  5  da.  at  4%. 

3.  $670  for  4  yr.  8  mo.  8  da.  at  4%%. 

4.  $950  for  6  yr.  6  mo.  10  da.  at  5%. 

5.  $500  for  8  yr.  7  mo.  11  da.  at  fy%. 

6.  $700  for  8  yr.  6  mo.  13  da.  at  6%. 

7.  $3000  for  5  yr.  8  mo.  14  da.  at  7%. 

8.  $600  for  4  yr.  9  mo.  16  da.  at  8%. 

9.  $300  for  4  yr.  8  mo.  17  da.  at  9%. 
10.  $536  for  3  yr.  10  mo.  19  da.  at  10%. 

The  Amount  equals  the  Principal  plus  the  Interest. 

1.  Find  the  amount  of: 

1.  $1  for  3  yr.  3  mo.  3  da.  at  6%. 

2.  $125  for  4  yr.  4  mo.  4  da.  at  6%. 


INTEEEST  253 

3.  $24.50  for  5  yr.  5  mo.  5  da.  at  7%. 

4.  $1000  for  6  yr.  6  mo.  6  da.  at  10%. 

5.  $280.75  for  7  yr.  7  mo.  7  da.  at  6%. 
2.  Find  the  interest  of: 

1.  $2000  for  5  moat 

2.  $6030  for  15  da.  at  4 

3.  $700  for  6  mo.  20  da.  at 

4.  $60.70  for  11  mo.  27  da.  at 

5.  $400  for  30  da.  at  6%. 

6.  $1670  from  April  1  to  Dec.  25  at  7%. 

7.  $4440  from  Feb.  4  to  Jime  8  at  5%. 

8.  $1060  from  April  13,  1897,  to  Dec.  21,  1898,  at 


Six  Per  Cent.  Method. 

MODEL   SOLUTION. 

At  6%  the  interest  of  $1.00  for  one  year  =  $.06  ;  for  one 
month  =  -JV  of  $.06  ==  $.OOJ  ;  for  one  day  =  -fa  of  $.OOJ 
=  $.0001  ' 

Hence,  writing  6  cents  for  every  year,  J  a  cent  for  every 
month,  and  J  of  a  mill  for  every  day,  we  have  the  formula  : 

/'  $.O6      x  yr.   } 

Int.  =  Pr.  X  \     -°°^     X  mo.  L  add. 
[    .POOj  X  da.  J 

What  is  the  interest  of  $236  for  3  yr.  4  rno.  18  da.,  at  6%  ? 

Process. 
r.18    } 

Int.  =  $236  X  1  .02     [  =  236  X  .203  =  $47.91. 
1  .003  J 

Explanation. 

For  3  yr.  we  write  $.18;  for  4  mo.,  $.02;  for  18  da.,  $.003  ;  the  sum 
of  these  three  is  $.203.  Since  the  int.  of  $1  is  $.203,  the  int.  of  $236  is 
236  times  $.203,  or  $47.91. 


254  PRACTICAL  ARITHMETIC 

EXERCISES. 

1.  Find  the  interest  of: 

1.  $560  for  3  yr. 

2.  $636  for  5  mo. 

3.  $700  for  60  da. 

4.  $236  for  1  yr.  10  mo.  18  da. 

5.  $35.60  for  4  yr.  9  mo.  24  da. 

6.  $2000  for  4  mo.  15  da. 

7.  $390.86  for  6  yr.  24  da. 

8.  $3000  for  5  yr.  11  mo.  25  da. 

9.  $6030  for  6  yr.  6  mo.  7  da. 
10.  $700  for  4  yr.  8  mo.  9  da. 

2.  Find  the  amount  of  : 

1.  $45.70  for  5  yr.  10  mo. 

2.  $443.76  for  9  mo.  12  da. 

3.  $1085.93  for  3  yr.  6  mo.  18  da. 

4.  $627.92  for  5  yr.  8  mo.  19  da. 

5.  $113.96  for  9  yr.  9  mo.  9  da. 

6.  $5090  for  10  yr.  10  mo.  10  da. 

7.  $3500  for  11  yr.  11  mo.  11  da. 

3.  Find  the  interest  of  $700  for  3  yr.  10  mo.  13  da.,  at 

Process. 
3yr.  =.18 

10  mo.  =  .05 
13  da.   =  .002| 

.232  £  X  700  =  162.517,  int.  at  6jg. 
6)162.517 

27.086,  int.  at  1%. 


$135.430,  int.  at  6%. 

NOTE.  —  Observe  how  the  six  per  cent,  method  is  applicable  when  other 
rates  are  given. 


INTEREST  255 

4.  Find  the  interest  of: 

1.  $760  for  3  yr.  11  mo.  12  da.,  at  5%. 

2.  $4030  for  5  yr.  3  mo.  7  da.,  at  7%. 

3.  $26.74  for  4  yr.  2  mo.  6  da.,  at  5|%. 

4.  $3000  for  6  yr.  6  mo.  6  da.,  at  4J%. 

5.  $2736  from  July  12,  1897,  to  Sept.  15,  1898,  at 

5%. 

6.  $526  from  Nov.  10,  1898,  to  June  16,  1900,  at 

7%. 

7.  $600  from  May  15,  1890,  to  July  11,  1898,  at 


EXACT   INTEREST. 

Exactness  requires  that  in  reckoning  interest  for  less  than 
one  year  365  days  should  be  considered  one  year,  and  not 

360  days.     Hence 

Pr.  X  R-  X  exact  No.  of  days 
Exact  Int.  =  —  —  365  — 

NOTE  1.  —  To  find  the  exact  number  of  days  between  two  dates  reckon 
each  year  as  365  days,  and  give  to  each  month  the  number  of  days  assigned 
it  in  the  calendar.  To  be  more  exact,  366  days  should  be  reckoned  for 
each  leap  year  and  29  days  to  February  of  every  leap  year,  but  in  the  fol- 
lowing problems  leap  years  are  not  considered. 

NOTE  2.  —  Since  5  days  —  -fa  of  365  days,  common  interest  diminished 
by  fa  of  itself  will  give  exact  interest  for  any  number  of  days  less  than  365. 
CAUTION.  —  This  does  not  apply  to  interest  reckoned  for  one  or  more  entire 
years. 

Find  the  exact  interest  of  $840,  at  6  %y  from  Mar.  3,  1894, 
to  Aug.  24,  1897. 

MODEL  SOLUTION. 
From  Mar.  3,  1894,  to  Mar.  3,  1897  =  3  yrs.  =  365  da.  X  3  =  1095  days 

Mar.    Apr.    May     June    July    Aug. 

From  Mar.  3,  1897,  to  Aug.  24,  1897=28+30+31+30+31+24=    174  days 

Exact  No.  of  days  =  1269  days 

168 

Exact  Int.  =  Pr'  X  K>  X  Exact  Na  of  days  •=  $&  X  .06  X  1269  =  $175  23 
365  30$ 

73 


256  PEACTICAL  ARITHMETIC 

EXERCISES. 

1.  Find  the  exact  interest  of: 

1.  $960  from  Feb.  5,  1898,  to  Dec.  26,  1898,  a 

2.  $2370  from  Apr.  10,  1887,  to  Aug.  15,  1890, 

at  5%. 

3.  $3500   from    Jan.  1,  1891,   to   Nov.  20,  1895, 

at  1%. 

4.  $2670  from  May  29,   1890,  to  Mar.  4,   1891, 

at  6%. 

5.  $4440  from  Feb.  5,  1898,  to  Dec.  25,  1898,  at  5  %  . 

2.  Find  the  exact  amount  of  : 

1.  $747.37  from  March  22,  1896,  to  Aug.  5,  1896, 

at  6%. 

2.  $837.46  from  April  3,  1896,  to  Dec.  21,  1897, 

at  9%. 

3.  $1094.94  from  Sept.  2,  1896,  to  Sept.  2,  1898,  at 

10%. 

4.  $231.03  from  Sept.  1,  1897,  to  Feb.  28,  1899, 

at  6%. 

5.  $556.44  from  Jan.  1,  1893,  to  April  21,  1900,  at 


To  Find  the  Principal,  the  Rate,  and  the  Time. 

FORMULAE. 
Since  Interest  =  Pr.  X  R  X  yr.,  obviously 

Int" 


1   Pr.  =  --          2.  R.  =  5-—.        3.  Yr.  =  p 

R.  X  yr.  Pr.  X  yr.  Pr.  X  R. 

Again,  since  Pr.  X  R.  X  yr.  +  Pr.  =  Amount,  and  since 
Pr.  X  R.  X  yr.  -|-  Pr.  =  Pr.  X  (R.  X  yr.  -f  1),  we  have, 
Pr.  X  (R.  X  yr.  -f  1)  =  Amount,  and 


4    Pr   -  Amt. 

4'  Pr>  ~  R.  X  yr.  +  1' 


INTEREST  257 

MODEL  SOLUTIONS. 

1.  What  principal  will  in  3  yr.  8  mo.  12  da.  yield  $1117.48 
interest  at  5%  ? 

Process. 

3  yr.    =  36     mo.  ^ 

8  mo.  =    8     mo.  V=  44.4  mo.  =3.7  yr.      Pr.  =        " 

xv-  A  j  x  • 
12  da.   =    0.4  mo.) 


2.  What  principal  will  in  4  yr.  7  mo.  6  da.  amount  to 
$859.52  at  4%  ? 

Process. 

4  yr.    =  48     mo.  ^ 

7  mo.  =    7     mo.  >  —55.2  mo.  =  4.6  yr.     Pr.  =  R  x  ™  '  ,  l 
6  da.   =    0.2  mo.  J 


.04  X  4.6  -f  1  ~ 

Or, 

$1.00  in  the  given  time  will  amount  to  .01  X  4  X  4.6  -f  1.00  —  $1.184. 
Since  $1.00  amounts  to  $1.184,  and  since  some  number  of  dollars  multiplied 
by  1.184  yields  $859.52,  that  number  must  be  $859.52  -=-  1.184,  or  $725.946. 

3.  At  what  rate  per  cent,  will  $4220  produce  $503.235 
interest  in  2  yr.  7  mo.  24  da.  ? 

Process. 

2  yr.    =  24     mo.  ^ 

7  mo.  =    7     mo.  >=  31.8  mo.  —  2.65  yr.     R.  =  pr  ^'  r 
24  da.  =    0.8  mo.  J 

503.235          _   503.235  _ 
4220  X  2.65  ~        11183      "  '°45  ==  4i%' 

4.  In  what  time  will  the  interest  on  $4220  at  4|%  amount 
to  $503.235? 

17 


258  PKACTICAL   ARITHMETIC 

Process. 
^T  Int.  503.235 

Yr-  =  P^TR:  =  42.20  X  4.5  "* 2-65  = 2  yr- 7  mo- 24  da- 

EXERCISES. 

1 .  Find  the  principal  that  will : 

1.  Produce  $180  int.  in  6  yr.  at  4%. 

2.  Produce  $126  int.  in  6  yr.  at  6J%. 

3.  Produce  $200  int.  in  16  yr.  6  mo.  at  5%. 

4.  Produce  $823.30  int.  in  1  yr.  11  mo.  at  6%. 
5;  Produce  $6  int.  in  14  mo.  at  5%. 

6.  Produce  $25  int.  in  144  da.  at  4J%. 

7.  Produce  $669.64  int.  in  1  yr.  7  mo.  12  da.  at  6%. 

8.  Produce  $2624.65  int.  in  2  yr.  6  mo.  at  5%. 

9.  Produce  $1680  in  6  yr.  at  4%. 

10.  Produce  $840  in  3  yr.  at  4J%. 

11.  Amount  to  $45,056.92  in  2  yr.  6  mo.  at  5%. 

12.  Amount  to  $3000  in  42  da.  at  5J%. 

13.  Amount  to  $595.20  in  16  mo.  at  6%. 

14.  Amount  to  $3189.375  in  2  yr.  2  mo.  at  5%. 

15.  Amount  to  $10,523.475  in  1  yr.  11  mo.  21  da. 

at  4J%. 

16.  Amount  to  $360.18  in  4  yr.  6  mo.  18  da.  at  5%. 

17.  Amount  to  $770.50  in  2  yr.  7  mo.  15  da.  at  6%. 

18.  Amount  to  $47,187.58  in  3  yr.  8  mo.  25  da.  at 


19.  Amount  to  $5133.30  in  4  yr.  6  mo.  27  da.  at  6%. 

20.  Amount  to  $950  in  3  yr.  3  mo.  3  da.  at  7%. 
2.  At  what  rate  will : 

1.  $1800  gain  $396  in  3  yr.  8  mo.? 

2.  $852  gain  $106.50  in  2  yr.  6  mo.? 

3.  $660  gain  $192.50  in  5  yr.  10  mo.? 

4.  $840  gain  $107,80  in  2  yr.  4  mo.? 


INTEREST  259 

5.  $144  gain  $128.52  in  12  yr.  9  mo.? 

6.  $220  gain  $82.36  in  3  yr.  8  mo.  ? 

7.  $420  gain  $42.30  in  2  yr.  9  mo.  24  da.  ? 

8.  $9.10  gain  $5.115  in  9  yr.  9  mo.  9  da.? 

9.  $100  double  itself  in  3  yr.  ?     5  yr.  ?     6  yr.  ? 

10.  Any  principal   treble  itself  in  7  yr.  ?     8  yr.  ? 

20  yr.? 
3.  Find  the  time  in  which  : 

1.  $500  will  produce  $60  interest  at  6%. 

2.  $1200  will  produce  $48  interest  at  8%. 

3.  $230  will  produce  $27.60  interest  at  6%. 

4.  $25.20  will  produce  $8.30  interest  at  7%. 

5.  $70.50  will  produce  $26.50  interest  at  7%. 

6.  $50  will  produce  $50  interest  at  6%. 

7.  $300  will  double  itself  at  8%. 

8.  $200  will  double  itself  at  5%.     6%.     7%. 

9.  Any  principal  will  double  itself  at  4J%. 

1 0.  Any  principal  will  treble  itself  at  6 % .    7%.    8%. 

PROBLEMS. 

1.  Find  the  exact  interest  of  $680.20,  at  7|%,  for  73  days. 

2.  What  sum,  bearing  interest  at  4^-%,  will  yield  an  annual 
income  of  $1500? 

3.  Find  the  amount  of  $1040  for  2  mo.  3  da.,  at  6%. 

4.  How  long  must  $1952.46  be  on    interest,  at  6%,  to 
amount  to  $2284.38  ? 

5.  At  what  rate  per  cent,  will  $6000  produce  $500  interest 
in  1  yr.  10  mo.  7  da.? 

COMPOUND    INTEREST. 

Compound  Interest  is  interest  computed,  at  certain  in- 
tervals, on  both  the  principal  and  unpaid  interest.  Such 
intervals  are  commonly  1  yr.,  6  mo.,  or  3  mo. 


260  PKACTICAL   ARITHMETIC 

MODEL   SOLUTIONS. 

1.  Find  the  amount  of  $70,  at  compound  interest  for  3  yr., 
at  6%  ;  also  the  compound  interest. 

Process. 

Int.  for  1st  yr.  =  Pr.  X  R.  X  yr.  =  70  X  .06  =  $4.20. 
Amt.  =  $74.20. 

Int.  for  2d  yr.  =  74.20  X  .06  =  $4.45.  Amt.  =  $78.65. 
Int.  for  3d  yr.  =  78.65  X  .06  =  $4.72.  Amt.  =  $83.37. 
(Amt.)  $83.37  —  (Pr.)  $70.00  =  $13.27,  compound  interest. 
Hence  the  formula : 

Compound  Int.  =  Final  Amount  —  Principal. 

2.  Find  the  compound  interest  of  $630  for  2  yr.  6  mo., 
at  5%. 

Process.  Explanation. 

Amt.  for  1st  yr.  =  $661.50.  2  yr  =  two  fu]]  intervalg .  , 

Amt,  for  2d   yr.  ==     694.58.  mo.  =  J  an  interval.     We  there- 

Ami  for  6  mo.    =     729.31.  fore  find  the  amount  of  $694.58 

$729.31  —  $630  ==  $99.31.          for  tbe  half  interva1'  6  m°' 

PROBLEMS. 

1.  Find  the  compound  interest  of  $200,  at  7%,  for  3  yr. 
6  mo. 

2.  What  is  the  amount  of  $458.50  for  2  yr.,  interest  com- 
pounded semi-annually,  at  6%  ? 

Suggestion  :  Compute  for  four  intervals  at  3%. 

3.  Compute  the  compound  interest  of  $580  for  1  yr.  3mo., 
interest  compounded  quarterly,  at  8%. 

Five  intervals,  1%. 

4.  Find  the  compound  interest,  at  6%,  on  $2000  for  1  yr. 
10  mo.,  interest  payable  semi-annually. 


INTEREST  261 

6.  What  is  the  compound  interest  of  $525.75  for  3  yr. 
4  mo.,  at6%? 

6.  Find  the  compound  interest  on  $1050  for  1  yr.  6  mo., 
at  5%,  interest  being  compounded  quarterly. 

7.  Compute  the  compound  interest  of  $600  for  2  yr.  3  mo., 
at  4%,  interest  being  compounded  semi-annually. 

8.  Find  the  compound  interest  of  $20,000  for  6  mo.,  at 
6%,  interest  being  compounded  monthly. 

ANNUAL   INTEREST. 

Annual  Interest  is  interest  on  the  principal  and  each 
year's  interest  from  the  time  each  interest  is  due  until  settle- 
ment. Annual  interest  is  computed  when  the  words  "  with 
interest  payable  annually  "  are  in  the  contract. 

MODEL  SOLUTION. 

Find  the  interest  of  $300  for  3  yr.  6  mo.  20  days  at  4%, 
payable  annually. 

Process. 

3  yr.  6  mo.  20  da.  =  1280  da. 
10     .01 

Int.  =  m  *j^-  =  $42.67,  for  the  whole  time. 

W 

3 

Int.  for  each  of  the  3  yr.  =  $12.00.  The  $12  -will  be  on 
interest : 

Firstly,  for  2  yr.  6  mo.  20  da. 
Secondly,  for  1  yr.  6  mo.  20  da. 
Thirdly,  for  6  mo.  20  da. 

4  yr.  8  mo.  =  total  time  =  56  mo. 

Int.  ==  &  x  -M  x  56  =  56  X  .04  ==  $2.24.      Total  Int.  == 
$42.67  +  $2.24  =  $44.91. 


262  PRACTICAL   ARITHMETIC 

Hence  the  following  brief  directions  : 

1.  Find  int.  of  Pr.  for  whole  time. 

2.  Find  int.  of  Pr.  for  one  yr. 

3.  Find  the  sum  of  the  time  intervals. 

4.  Find  int.  on  the  one  year's  int.,  for  the  sum  of  the 
time  intervals. 

5.  Find  the  sum  of  int.  first  found  and  int.  last  found. 

EXERCISES. 

1.  Find  the  annual  interest  of: 

1.  $360  for  4  yr.  5  mo.  16  da.  at  6%. 

2.  $250  for  3  yr.  9  mo.  12  da.  at  1%. 

3.  $3500  for  4  yr.  6  mo.  at  6%. 

4.  $1247.75  for  3  yr.  5  mo.  10  da.  at  6%. 

5.  $987.25  for  4  yr.  9  mo.  6  da.  at  4%. 

6.  $1098.36  for  5  yr.  10  mo.  7  da.,  at  5%. 

2.  Find  the  amount,  at  annual  interest,  of: 

1.  $360  for  4  yr.  5  mo.  16  da.  at  5%. 

2.  $250  for  3  yr.  9  mo.  12  da.  at  7%. 

3.  $600  for  3  yr.  4  mo.  12  da.  at  6%. 

4.  $840  for  4  yr.  8  mo.  18  da.  at  5J%. 

5.  $2180  for  6  yr.  11  mo.  27  da.,  at  4J#. 

6.  $1070  for  5  yr.  10  mo.  24  da.  at  4%,  the  interest 

of  the  first  two  years  having  been  paid. 

PROMISSORY   NOTES. 

1.  A  Promissory  Note  is  a  promise,  made  in  writing,  to 
^ay  a  sum  of  money  091  demand  or  at  a  specified  time. 

2.  The  Face  of  a  note  is  the  sum  of  money  named  in  it. 

3.  The  Maker  of  a  note  signs  it.     The  Payee  receives 
payment  for  it.     The  Holder  has  rightful  possession  of  it. 

4.  The  Endorser  of  a  note  writes  his  name  on  the  back 
of  it,  and  thus  becomes  responsible  for  payment  of  it. 


PROMISSORY  NOTES  263 

5.  A  Negotiable  Note  is  one  that  is  transferable. 

6.  Notes  are  said  to  be  negotiable  or  transferable  when 
they  contain  the  words  "  or  bearer/'  or  "  or  order/7  but  no 
transfer  of  the  latter  can  be  made  without  the  endorsement  of 
the  payee. 

To  insure  the  negotiability  of  a  note,  in  Pennsylvania  the 
words  "  without  defalcation"  should  be  added.  In  New  Jersey 
the  words  "  without  defalcation  or  discount'7  should  be  added  ; 
in  Missouri,  "negotiable  and  payable  without  defalcation  or 
discount." 

7.  The  words  "with  interest"  render  the  note  interest- 
bearing  from  its  date. 

8.  A  note  not  containing  the  words  "  with  interest"  begins 
to  bear  interest  at  maturity  if  not  paid. 

9.  The  words  "  value  received"  are  proof  that  the  note 
represents  actual  value. 

10.  The  day  of  maturity  of  a  note  is  the  day  when  it  be- 
comes due. 

11.  In  any  case,  when  the  rate  per  cent,  is  not  specified  the 
legal  rate  is  always  understood. 

12.  Interest  computed  at  a  higher  rate  than  the  law  allows 
is  called  usury. 

13.  In  many  States  the  time  of  payment  is  postponed  three 
days,  called  "  Days  of  Grace." 

14.  A  Protest  is  a  notice  sent  to  the  endorsers  that  the 
maker  of  the  note  has  failed  to  pay  it.     The  protest,  to  be 
valid,  must  not  be  sent  later  than  the  last  day  of  grace. 

15.  A  note  signed  by  two  or  more  persons,  who  thus  become 
jointly  and  severally  responsible  for  its  payment,  is  called  a 
Joint  or  Several  Note. 


264  PRACTICAL   ARITHMETIC 

v 

Forms  of  Notes. 

PO 

CHICAGO,  ILL.,  Sept.  1,  1898. 

Three  months  after  date,  I  promise  to  pay 
Edward  L.  Baker- 
Four  hundred  eighty-six-—  -  -  —  -  ----------  jSUL.  Dollars, 

with  interest  at  5%,  for  value  received. 

EGBERT  H.  KING. 


WASHINGTON,  D.  C.,  Sept.  1,  1898. 

Four  months  after  date,  I  promise  to  pay 
Edward,  L.  Baker  —  -----  —  —  or  bearer, 

Four  hundred  eighty-six  -       ~—  -----------------  ^-f^  Dollars, 

with  interest  at  7%,  for  value  received. 

ROBERT  H.  KING. 

(3.) 

$486^ 

PHILADELPHIA,  PA.,  Sept.  1,  1898. 

Six  months  after  date,  /  promise  to  pay 
Edward  L.  Baker  —  -  ^^~^—  __________  ^r  order, 

Four  hundred  eighty-six^-  -  ------  ~-f-f^  Dollars, 

without  defalcation,  for  value  received. 

ROBERT  H.  KING. 

(*•) 

$486^ 

TRENTON,  N.  J.,  Sept.  1,  1898. 

On  demand,  I  promise  to  pay  Edward  L.  Baker 
Four  hundred  eighty-six  -  --------------------  ffa  Dollars, 

with  interest  at  6%,  without  defalcation  or  discount. 

ROBERT  H.  KING. 


PROMISSORY  NOTES  265 

(5.) 

$486^ 

ST.  Louis,  Mo.,  Sept.  1,  1898. 

Four  months  after  date,  we  jointly  and  severally 
promise  to  pay  Edward  L.  Baker—-  —or  order, 

Four  hundred  eighty-six — ff$  Dollars, 

with  interest  at  3%,  for  value  received,  negotiable  and  payable 
without  defalcation  or  discount. 

ROBERT  H.  KING. 

JOHN  C.  TAYLOR. 

(6.) 

ATLANTA,  GA.,  Sept.  1,  1898. 
Sixty  days  after  date,  I  promise  to  pay 

Edward  L.  Baker- — —or  order, 

at  the  Atlanta  National  Bank 

Four  hundred  eighty-six-^  ^"ffo  Dollars, 

with  interest,  value  received. 

ROBERT  H.  KING. 


QUESTIONS  AND  EXERCISES. 

1.  Find  when  the  above  notes  will  severally  mature. 

2.  Compute  the  amount  due  on  each  at  maturity. 

3.  Point    out    which    are    negotiable    and    which    non- 
negotiable. 

4.  Point  out  which  are  interest-bearing  from  date  and 
which  from  maturity. 

5.  When  may  a  "  demand"  note  be  collected  ? 

6.  When  is  a  time  note  collectible  ? 

7.  When  no  rate  per  cent,  is  specified  in  a  note,  what  rate 
is  understood  ? 

8.  What  is  the  legal  rate  in  your  State  ? 

9.  Can  a  note  be  protested  after  its  maturity  ? 


266  PRACTICAL  ARITHMETIC 

10.  If  the  maker  of  a  note  fails  to  pay  it,  who  is  held 
responsible  for  payment  of  it? 

11.  Write  a  negotiable  note,  in  favor  of  George  Hudson, 
for  $500.50,  using  your  own  name  as  that  of  maker. 

12.  Write  a  note  from  the  following  data:  Face,  $347.56  ; 
negotiable ;    time,   60  days ;    payee,  George  Jones ;   maker, 
Hiram  Smith;  rate,  6%  ;  place,  Reading,  Pa. 

13.  Write  a  non-negotiable  note. 

14.  Write  a  note  that  will  bear  interest  from  date. 

15.  Write  a  note  payable  at  a  bank. 

16.  Write  a  note  with  an  endorsement. 

NOTE. — Latin,  dorsum,  the  back. 

17.  Write  a  note  payable  with  annual  interest. 

18.  Assume  a  date  for  settlement,  and  compute  the  amt. 
due  on  said  note. 

19.  Find  the  day  of  maturity  and  amount  due,  having 
given  : 

1.  $631.36,  Feb.  13,  1898,  63  da.,  6%. 

2.  $796.56,  Apr.  23,  1898,  90  da.,  5%. 

3.  $397.86,  Sept.  6,  1898,  5  mo.,  4%. 

4.  $1055.51,  Nov.  21,  1898,  4  mo.,  6%. 

5.  $631.36,  Nov.  6,  1898,  33  da.,  7%. 

6.  $937.72,  Jan.  17,  1898,  6  mo.,  8%. 

7.  $2632.98,  Apr.  30,  1898,  1  mo.,  10%. 

8.  $2849.65,  June  23,  1898,  15  da.,  6%. 

9.  $984.05,  Aug.  11,  1898,  3  yr., 
10.  $1968.10,  Sept.  3,  1898,  3  mo., 

PARTIAL   PAYMENTS. 

1.  The  payment  of  part  of  a  note  or  other  obligation  is 
called  a  Partial  Payment. 

2.  Notes  on  which  partial  payments  have  been  endorsed  are 


PARTIAL  PAYMENTS  267 

computed  chiefly  by  two  rules :  The  Merchants'  Rule  and 
The  United  States  Rule. 

The  Merchants'  Rule. 

The  Merchants'  Rule  applies  to  notes  settled  within  a  year. 
The  method  is  as  follows  : 

MODEL   SOLUTION. 

PITTSBURO,  PA.,  Feb.  25,  1898 
For  value  received,  I  promise  to  pay  John 
Wayland,  or  order,  Six  Hundred  Dollars,  on  demand,  with 
interest  from  date. 

JAMES  BROWN. 

On  this  note  were  made  the  following  payments :  May  25, 
1898,  $156.00;  Aug.  25,  1898,  $200.00;  Nov.  25,  1898, 
$185.00.  What  was  due  on  Feb.  20,  1899  ? 

Process. 

Date  of  settlement,  1899,  2,  20. 
Date  of  note,  1898,  2,  25. 

Interval,  11  mo.  25  da. 

Principal  (note) $600.00. 

Interest  for  11  mo.  25  da.  35.50. 


Amount $635.50. 

1899,  2,  20.         1899,  2,  20.         1899,  2,  20. 
1898.  5,  25.         1898,  8.  25.         1898,  11,  25. 

8  mo.  25  da.        5  mo.  25  da.        2  mo.  25  da. 

1st  payment  and  int.  for  8  mo.  25  da $162.89. 

2d  payment  and  int.  for  5  mo.  25  da 205.84. 

3d  payment  and  int.  for  2  mo.  25  da 187.62.        556.35. 

Balance  due  at  settlement $79.15. 

Hence  the  formula : 

Balance  =  Amount  of  Pace  due  at  time  of  settlement  — 
the  sum  of  the  Payment- Amounts  due  at  time  of  settle- 
ment. 


268  PRACTICAL  ARITHMETIC 

PROBLEMS. 

1.  A  note  for  $960,  on  demand,  with  interest  at  7%,  dated 
Feb.  1,  1897,  was  endorsed  as  follows :  May  11,  1897,  $300 ; 
Oct.  16,  1897,  $366.     How  much  was  due  Dec.  16,  1897? 

2.  What  amount  is  due  Nov.  27, 1898,  on  a  note  for  $800, 
dated  Jan.  15,  1898,  with  interest  at  6%,  on  which  are  the 
following  endorsements:  May  3,  1898,  $300;  July  9,  1898, 
$400? 

3.  A  man  holds  a  note  of  $460,  dated  Jan.  20,  1898,  on 
which  the  following  payments  are  endorsed :  $140,  Mar.  26, 
1898;  $100,  June  16,  1898;  $160,  Oct.  14,  1898.     Settle- 
ment is  made  Dec.  22,  1898.     Find  the  balance  due,  interest 
at  5%. 

4.  What  is  due  Dec.  20,  1898,  on  a  note  for  $1300,  dated 
Feb.  10,  1898,  with  interest  at  fy%,  on  which  is  the  following 
endorsement:  June  7,  1898,  $900? 

5.  A  note  of  $1100,  dated   April   1,  1898,  payable  on 
demand,  with   interest  at  6%,  bears  the  following  endorse- 
ments :   June  6,  $300;   Aug.    5,   236.48;   Nov.    19,  $333. 
What  is  due  Jan.  1,  1899? 

(6.) 

$696^ 

BIRMINGHAM,  ALA.,  April  4,  1898. 

Nine  months  after  date,  for  value  received, 
I  promise  to  pay  to  the  order  of  Paul  Stakeman,  Six  Hundred 
Ninety-six  and^^-  Dollars,  with  interest  at  8%. 

ROBERT  S.  CAMPBELL. 

This  note  was  endorsed  as  follows:  July  9,  1898,  $436; 
Sept.  4,  1898,  $95.40;  Oct.  3,  1898,  $100.  What  was  due 
on  the  note  at  maturity  ? 

7.  A  note  of  $946.36,  dated  Aug.  1,  1898,  payable  on 
demand,  with  interest  at  5^%,  bears  the  following  endorse- 


PARTIAL  PAYMENTS  269 

ments :  Sept,  21,  $268.60 ;  Oct.  22,  $280.36 ;  Nov.  6,  $300 ; 
Dec.  2,  $90.     What  remained  due  after  Dec.  2  ? 

8.  A  note  for  $4300,  dated  Feb.  8,  1898,  has  the  fol- 
lowing endorsements  on  it :  Mar.  20,  1898,  $900 ;  April  20, 

1898,  $700;   July  25,  1898,  $600;   Aug.  17,  1898,  $500; 
Nov.  25,  1898,  $400.     What  is  due  Jan.  1,  1899,  at  6%  ? 

(9.) 

$9600-,%- 

HARRISBTJRG,  PA.,  July  20,  1898. 

Thirty  *  days  after  date,  for  value  received, 
I  promise  to  pay  Charles  Davenport,  or  order,  Nine  Thousand 
Six  Hundred  Dollars,  without  defalcation. 

JAMES  H.  BOYD. 

Endorsements:  Aug.  30,  $200;  Oct.  12,  $400;  Nov.  10, 
$600  ;  Dec.  20,  $800.     What  is  due  July  15,  1899  ? 

10.  A  note  for  $7000,  at  90  days,  dated  Sept.  25,  1898, 
has  the  following  endorsements :   Dec.   31,  $300 ;   Jan.  18, 

1899,  $500;  May  20,  1899,  $600;  Aug.  10,  $100.     What 
was  due  Sept.  1,  1899? 

11.  On  a  note  for  $2898,  dated  Jan.  4,  1897,  and  bearing 
interest  at  5^%,  the  following  payments  were  made  :  Jan.  20, 
1897,  $600;   Feb.  25,  1897,  $700;   June  10,  1897,  $400; 
Sept.  30, 1897,  $360.    How  much  is  due  Jan.  1,  1898. 

The  United  States  Rule. 

This  rule  applies  to  notes  settled  beyond  the  limit  of  a  year, 
and  forbids  the  deducting  of  a  payment  unless  the  payment 
equals  or  exceeds  the  interest  due.  The  compounding  of  in- 
terest is  thus  prevented. 

*  Interest  must  not  be  computed  for  these  30  days. 


270  PRACTICAL  ARITHMETIC 

MODEL   SOLUTION. 


NEW  YORK,  May  1,  1895. 

I  promise  to  pay  George  Jenkins,  or  order, 
Four  Hundred  Seventy-five^-  Dollars,  on  demand,  with 
interest  at  7%,  for  value  received. 

THOMAS  WRIGHT. 

Endorsements:  Dec.  25,  1895,  $50.00;  July  10,  1896, 
$15.75;  Sept.  1,  1897,  $25.50;  June  14,  1898,  $104.00. 
How  much  is  due  April  15,  1899? 

Process. 
Face  of  note    .......................    $475.50. 

Date  of  1st  pay't,  1895,  12,  25. 
Date  of  note,  1895,     5,     1. 

Interval  7  mo.  24  da. 

Int.  of  face  for  interval     ..................        21.64. 

Amount    .............    $497.14. 

1st  payment  (exceeding  interest)     ..............        5000. 

1st  balance   ............    $447.14. 

As  the  next  two  payments  will  not  equal  or  exceed  the  interest 
due,  we  compute  the  interval  to  June  14,  1898. 
Date  of  4th  pay't,  1898,     6,  14. 
Date  of  1st  pay't,    1895,  12,  25. 

Interval,  2  yr.  5  mo.  19  da. 

Int.  of  1st  balance  for  interval    ...............        77.29. 

Amount    .............    $524.43. 

2d  payment  ................    $15.75 

3d  payment  ................      25.50 

Sum  less  than  int.  due   ..........    $41.25 

4th  payment    ...............    104.00       14525. 

2d  balance    ......    ......    $379.18. 

Date  of  settlement,   1899,  4,  15. 
Date  of  4th  pay't,     1898,  6,  14. 

Interval,  10  mo.  1  da. 

Int,  of  2d  bal.  for  interval   .................        22.19. 

Balance  due  April  15,  1899  .....    $401.37. 


PARTIAL  PAYMENTS  271 

BULB. 

1.  Find  the  amount  of  the  principal  to  the  given  date 
at  which  the  payment,  or  sum  of  the  payments,  is  equal 
to  or  greater  than  the  interest. 

2.  From  this  amount  deduct  the  payment,  or  the  sum  of 
the  payments. 

3.  Consider  the  remainder  as  a  new  principal  and  pro- 
ceed as  before. 

PROBLEMS. 

(!•) 

$3000^ 

PHILADELPHIA,  PA.,  Feb.  26,  1895. 

On  demand,  I  promise  to  pay  George  Palmer 
Three  Thousand  Dollars,  with  6%  interest. 

JOHN  JAY. 

Payments  were  made  on  this  note  as  follows:  Sept.  10, 
1895,  $25.00;  Jan.  1, 1896,  $500.00;  Oct.  25,  1896,  $75.00; 
April  4, 1897,  $1500.00.  How  much  was  due  Feb.  20, 1898  ? 

(20 

$750T% 

LYNN,  MASS.,  July  15,  1893. 

Three  months  after  date,  I  promise  to  pay 
George  Mason,  or  order,  Seven  Hundred  and  Fifty  Dollars, 
for  value  received. 

GEORGE  PALMER. 

Payments  were  made  as  follows:  Aug.  3,  1896,  $75.00; 
May  1,  1897,  $560.00.  What  was  due  Feb.  20,  1898  ? 

3.  A  note  was  given  Jan.  1,  1890,  for  $700.    The  follow- 
ing payments  were  endorsed  upon  it:   May  6,  1890,  $85; 
July   1,  1891,  $40;  Aug.  20,  1891,  $100;  Jan.  10,  1893, 
$350.     How  much  was  due  Sept.  30,  1894,  interest  at  6%  ? 

4.  What  is  due  Aug.  18,  1894,  on  a  note  for  $400,  dated 
April  1,  1892,  with  interest  at  6%,  on  which  are  the  following 


272  PRACTICAL  ARITHMETIC 

endorsements  :  Jan.  13, 1893,  $50 ;  Sept.  22, 1893,  $10  ;  April 
25,  1894,  $125. 

(5.) 

$750^0- 

PHILADELPHIA,  PA.,  May  10,  1895. 

Three  years  after  date,  for  value  received, 
I  promise  to  pay  Thomas  Newbury,  or  order,  Seven  Hundred 
and  Fifty  Dollars,  with  interest,  without  defalcation. 

SAMUEL  TOWNSEND. 

Endorsements:  Jan.  15,  1896,  $124.75;  Sept.  12,  1896, 
$20;  Dec.  16,  1897,  $216.80.  How  much  remained  due 
May  10,  1898? 

6.  A  note  for  $5600  was  given  May  1,  1895,  and  was 
endorsed  as  follows:  Oct.  17,  1895,  $350;  Feb.  18,  1896, 
$455  ;  July  10, 1896,  $318.50.     What  was  due  May  1, 1897, 
interest  at  1%  ? 

7.  What  is  due  Dec.  31,  1898,  on  a  note  for  $2800,  dated 
Oct.  10,  1894,  with  interest  at  7%,  on  which  are  the  follow- 
ing endorsements:   April  1,  1895,  $66.60;  July  21,  1896, 
$300  ;  June  15,  1898,  $300  ? 

(8.) 
$4600-^ 

CHICAGO,  ILL,  April  6,  1896. 

On  demand,  I  promise  to  pay  George  K.  Brown, 
or  order,  Four  Thousand  Six  Hundred  Dollars,  for  value  re- 
ceived, with  interest  at  5  % . 

CHARLES  MOREHEAD. 

Endorsements  :  July  10, 1897,  $1360  ;  Oct.  4, 1897,  $500  ; 
Jan.  16,  1898,  $660 ;  June  21,  1898,  $700.  How  much  was 
due  Jan.  21,  1899? 

9.  A  note  for  $8580  was  given  July  12, 1894.  Endorsed  : 
Jan.  8,  1895,  $300;  April  26,  1896,  $500;  July  16,  1897, 


BANK  DISCOUNT  273 

$335;   Oct.   8,  1897,  $250.     What  was  due  at  settlement, 
Jan.  1,  1898? 

10.  A  note  was  drawn  in  Michigan  for  $2774.65,  payable 
in  2  years,  with  interest,  and  dated  March  15,  1896.  Pay- 
ments were  made  as  follows:  July  30,  1896,  $100;  Dec.  8, 
1896,  $200;  Jan.  5,  1897,  $250;  May  17,  1897,  $600;  Jan. 
1,  1898,  $600.  How  much  remained  unpaid,  April  1,  1898? 

BANK  DISCOUNT. 

1.  A  Bank  is  an  institution  established  for  the  purpose 
of  receiving,  loaning,  and  issuing  money. 

NOTE. — All  banks  do  not  issue  money. 

2.  For  cashing  notes  in  advance  of  their  maturity,  banks 
make  a  deduction  from  their  face  value.     This  deduction  is 
called  Bank  Discount. 

3.  Bank  discount  depends  upon  Face,  Rate,  and  Time,  and 
is  computed  precisely  like  simple  interest. 

4.  The  Time  of  Discount  of  a  note  is  the  interval  between 
the  day  of  its  presentation  and  the  day  of  its  maturity.     This 
interval  is  commonly  called  time  to  run. 

NOTE. — In  some  States  the  time  to  run  is  increased  by  3  days,  called 
"Days  of  Grace." 

5.  The  Proceeds  of  a  note  equal  its  Face  less  the  Discount. 

MODEL   SOLUTION. 

$2360^ 

PHILADELPHIA,  PA.,  Feb.  26,  1897. 

Three  months  from  date,  I  promise  to  pay  to 
the  order  of  George  Gross,  at  the  West  Philadelphia  Bank, 
Twenty-three  Hundred  and  Sixty  Dollars,  for  value  received. 

JAMES  JENKINS. 


274  PRACTICAL  ARITHMETIC 

This  note  was  presented  at  bank  for  discount  April  1, 1897. 
Find  :  1.  The  day  of  maturity.  2.  The  time  to  run.  3.  The 
discount.  4.  The  proceeds. 

Process. 

Feb.  26,  1897  +  3  mo.  =  May  26, 1897,  the  day  of  maturity. 
Day  of  maturity,        1897     5     26 
Day  of  presentation,  1897     4       1 

1  mo.  25  da.,  time  to  run. 
rv  x  r       cr  j  236°  X  -06  X  55         A  01  nA 

Discount  for  55  da.  =  -        36Q        -  =  $21.64. 

Face  of  note  =  $2360.00 

Discount        =        21.64 

Proceeds         =  $2338.36 

That  is,  the  bank  took  the  note  and  paid  in  cash  for  it 
$2338.36. 

Hence  the  brief  directions  are  : 

1.  Find  the  day  of  maturity. 

2.  Find  the  time  to  run. 

3.  Find  the  discount  (simple  interest). 

4.  Find  the  proceeds  (subtract  discount  from  face). 

EXERCISES. 

Find  the  discount  and  proceeds  of: 

1.  $350  for  30  da.  at  5%. 

2.  $400  for  90  da.  at  6%. 

3.  $540  for  60  da.  at  7%. 

4.  $600  for  60  da.  at  8%. 

5.  $2000  for  3  mo.  at  10%. 

6.  $80.60  for  90  da.  at  5J%. 

7.  $5000  for  18  da.  at  6J%. 

8.  $780  for  40  da.  at  7J%,  with  grace. 

9.  $600  for  2  mo.  12  da.  at  8J%,  with  grace. 
10.  $1000  for  90  da.  at  10%,  with  grace. 


BANK  DISCOUNT  275 

PROBLEMS. 
Apply  the  brief  directions  to  the  following  notes  : 

00 

$5003% 

SAN  FRANCISCO,  CAL.,  Feb.  20,  1898. 
Sixty  days  after  date,  I  promise  to  pay 

James   Warner,    or    order,    Five    Hundred    Dollars,    value 
received.  JOHN  GORDON. 

Discounted  Mar.  15,  1898,  at  7%. 

(2.) 

$800^ 

BALTIMORE,  MD.,  Feb.  1,  1898. 

Ninety  days  after  date,  I  promise  to  pay  to  the 
order  of  Peter  Welsh  Eight  Hundred  Dollars,  for  value 
received.  HENRY  BRYCE. 

Discounted  April  1,  1898,  at  6%. 

(3.) 


PHILADELPHIA,  PA.,  Jan.  5,  1898. 
Ninety  days  after  date,  I  promise  to  pay 
Charles   Garrett,  or  order,   Four   Hundred   Dollars   at   the 
Girard  Bank,  for  value  received,  without  defalcation. 

JOHN  WATERMAN. 
Discounted  Jan.  10  at  6%. 

(4-) 


WASHINGTON,  D.  C.,  April  20,  1898. 
Six  months  after  date,  for  value  received, 
I  promise  to  pay  Alfred  Rickert,  or  order,  Four  Hundred 
Sixty-five12^j-  Dollars,  at  the  First  National  Bank. 

WESLEY  EVANS. 
Discounted  June  23  at  7%. 


276  PRACTICAL  ARITHMETIC 

5.  Find  the  bank  discount  of  a  note  for  $3600,  dated  March 
6,  1898,  and  payable  3  mo.  after  date,  with  interest  at  5%,  if 
discounted  May  13,  1898,  at  6%. 

6.  Find  the  proceeds  of  a  note  for  $2400,  dated  Aug.  26, 
1898,  payable  90  days  after  date,  with  interest,  at  5J%,  and 
discounted  Oct.  1,  1898,  at  6%. 

To  find  the  Face  of  a  Note. 

It  is  sometimes  necessary  to  determine  what  face  to  give  a 
note  in  order  to  secure  a  certain  sum  as  proceeds. 

Find  the  face  of  a  note  that,  discounted  for  60  days  at  6%, 
will  yield  $500  as  proceeds. 

Process.  Explanation. 

Discount  of  $1  .00  =  .01  .  Since  S1-00'  as  face> 

Proceeds  of  $1.00  -  1.00  -  .01  =  .99.  Counted  for  60  da. 


$500,-.  99  =  $505.05. 

many  dollars  as  face  will  yield  $500  as  proceeds  ?     Obviously,  $500  -=-  .99 
=  $505.05. 

Hence  the  formula  : 

Face  =  Given  Proceeds  .-=-  Proceeds  of  $1.OO. 
Face  =  Given  Discount  •*•  Discount  of  $1.OO. 

EXERCISES. 

Find  the  face  in  each  of  the  following  instances  : 

1.  Proceeds,  $800;  time  60  da.  ;  rate,  6%. 

2.  Proceeds,  $989.50;  time,  2  mo.;  rate,  6%. 

3.  Proceeds,  $3000  ;  time,  90  da.  ;  rate,  6%. 

4.  Proceeds,  $15,000;  time,  2  mo.;  rate,  7%.     Grace. 

5.  Proceeds,  $240;  time,  3  mo.;  rate,  5%.     Grace. 

6.  Proceeds,  $975;  time,  2  mo.;  rate,  1%.     Grace. 

7.  Discount,  $40;  time,  90  da.  ;  rate,  6%. 

8.  Discount,  $4.18  ;  time,  60  da.  ;  rate,  6%. 


TRUE  DISCOUNT  277 

9.  Discount,  $8.48;  time,  60  da. ;  rate,  5%. 
10.  Discount  $17.50;  time,  2  mo.  12  da.;  rate,  7%. 

PROBLEMS. 

1.  I  wish  to  borrow  $400  at  a  bank.     For  what  sum  must 
I  draw  my  note,  payable  in  60  da.,  so  that  when  discounted 
at  6%  I  shall  receive  the  desired  sum  ? 

2.  What  is  the  face  of  a  note  at  60  days  which  yields  $780 
when  discounted  at  a  bank?     Rate,  5%. 

3.  Suppose  you  buy  goods  in  Philadelphia  to  the  amount 
of  $1248.50,  and  give  your  note  in  payment  drawn  at  6  mo. 
What  must  be  the  face  of  the  note  ? 

4.  For  how  large  a  sum  must  a  note  be  drawn,  payable  in 
3  mo.,  that  the  net  proceeds  may  be  $7500  after  deducting 
the  bank  discount  at  8%  ? 

5.  A  Chicago  merchant  sold  goods,  and  received  in  payment 
for  them  a  6-mo.  note,  which  he  had  immediately  discounted 
at  7%.     If  he  received  $1898  in  cash  for  the  note,  for  what 
sum  had  he  sold  the  goods  ? 

6.  For  what  amount  must  a  note  be  made  payable  in  3  mo., 
so  that  when  discounted  in  Baltimore  at  the  legal  rate  (6%), 
the  proceeds  may  be  $1420. 

7.  For  what  sum  must  a  note  be  drawn,  payable  in  3  mo., 
so  that  when  discounted  in  Montana  at  the  legal  rate  (10%), 
the  proceeds  may  be  $1000? 

8.  In  Oregon  I  suffered  a  discount  of  $6.18  on  a  6-mo. 
note  at  the  legal  rate  (8%).     Find  the  face  of  my  note? 

TRUE  DISCOUNT. 

1.  The  Present  "Worth  of  a  debt  is  a  sum  which,  put  at 
interest,  amounts  to  the  debt  when  due. 

2.  True  Discount  is  the  difference  between  the  present  worth 
and  the  debt.    Finding  the  present  worth  is  the  same  as  finding 


278  PRACTICAL  ARITHMETIC 

what  principal  will  in  a  given  time,  and  at  a  given  rate, 
amount  to  a  given  sum. 

Hence  we  have,  from  page  256,  Pr.  —  R  x  ™tj  ,  1?  which 

becomes : 

T->  TTT         Amt.  or  Debt 
RW'=R.  xyr.  +  17 

MODEL  SOLUTION. 

What  present  worth,  or  principal,  will  amount  to  $1000  in 
8  mo.  at  6  %  ?     Also,  find  the  true  discount. 

Process. 

p    W  Amt.  HOOP          =  $1000  =  ftofi-i   54 

~  K.  X  yr.  +  1         .06  X  f  +  1       '    1-04    " 

$1000  —  $961.54  =  $38.46,  true  discount. 

Explanation. 

Since  the  P.  W.  stands  to  the  Amt.  in  the  relation  of  principal,  we  use 
the  formula,  P.  W.  =  R  x^+  1?  and  obtain  $961.54.     $1000  —  $961.54 

=  $38.46,  true  discount. 

Or,  we  may  say  :  $1.00  amounts  to  $1.04 ;  therefore,  it  will  require  the 
quotient  of  $1000  -=-  1.04  to  amount  to  $1000.    Hence  the  P.  W.  =  $961.54. 

EXERCISES. 

Find  the  present  worth  and  true  discount  of: 

1.  $400,  due  1  yr.  hence,  at  6%. 

2.  $200,  due  1J  yr.  hence,  at  6%. 

3.  $180,  due  1  yr.  5  mo.  hence,  at  5%. 

4.  $600,  due  2  yr.  3  mo.  hence,  at  8%. 

5.  $350,  due  2  yr.  6  mo.  9  da.  hence,  at  7%. 

6.  $1500,  due  2  mo.  21  da.  hence,  at  6%. 

7.  $2000,  due  2  yr.  3  mo.  6  da.  hence,  at  6%. 

8.  $487.75,  due  3  yr.  hence,  at  7%. 

9.  $422.00,  due  2J  yr.  hence,  at  6%. 
10.  $479.37 J,  due  3  yr.  hence,  at  5%. 


T£UE  DISCOUNT  279 

PROBLEMS. 

1.  Find  the  present  worth  and  true  discount  of  $200,  due 
in  3  yr.  8  mo.  16  da.,  at  5J%. 

2.  Find  the  bank  discount  on  $1000,  due  in  9  mo.,  without 
grace,  money  being  worth  6%. 

3.  Find  the  difference  between  the  bank  discount  and  the 
true  discount  of  $1000,  due  in  9  mo.,  rate  5%. 

4.  Find  the  true  discount  on  $980,  due  in  6  mo.,  money 
being  worth  4J%. 

5.  Money  being  worth  6%,  find  the  difference  between  the 
true  discount  and  the  bank  discount  of  a  note  for  $525,  due 
in  10  mo.,  without  interest. 

6.  If  I  buy  goods  for  $3000  on  3  mo.  credit,  what  dis- 
count should  I  receive  if  I  pay  cash,  money  being  worth  5  J  %  ? 

7.  If  I  pay  a  debt  of  $9450  2  yr.  6  mo.  15  da.  before  it 
is  due,  what  discount  should  I  receive,  money  being  worth 
8%? 

8.  What  is  the  aggregate  present  worth  of  two  notes,  each 
for  $800,  due  at  the  end  of  one  and  three  years  respectively, 
the  rate  of  bank  discount  being  7%  ? 

9.  I  wish  to  place  at  6%  interest  a  sum  that  will  amount 
to  $832.50,  from  Jan.  9,  1897,  to  Nov.  9,  1898.    What  is  the 
sum? 

10.  If  you  owe  $500,  to  be  paid  in  1  yr.,  without  interest, 
what  ought  you  in  equity  to  pay  now  in  order  to  cancel  the 
debt,  if  money  is  worth  7  %  ? 

REVIEW. 

1.  On  property  worth  $15,000,  fire  caused  a  loss  of  $3840. 
Find  the  rate  per  cent,  of  loss. 

2.  An  agent  makes  20%  by  selling  a  book  for  $2.88. 
Had  he  sold  it  for  $4.00,  what  per  cent,  would  he  have  made? 


280  PRACTICAL  ARITHMETIC 

3.  Find  the  interest  on  $960  for  7  yr.  6  mo.  27  da.,  at 
4J%.   ^Also,  find  the  interest  at  9%. 

4.  Find    the   rate   per    cent,    when    $1758    amounts    to 
$1869.34  in  8  mo. 

5.  Find  the  time  when  the  principal,  at  7%,  is  doubled. 

6.  What  principal  will  amount  to  $2222.22  in  2  yr.  2  mo. 
2  da.,  at  5%?   • 

7.  The  face  of  a  note  is  $1975 ;  its  date,  Sept.  12,  1898  ; 
its  time,  3  mo. ;  its  day  of  discount,  Sept,  26,  1 898 ;  its  rate 
of  discount,  5J%.     Find  its  day  of  maturity,  etc. 

8.  Find  the  compound  interest  of  $4000  for  2  yr.  6  mo., 
at  5%  per  annum. 

9.  Find  the  annual  interest  of  $1600  for  4  yr.  8  mo.,  at 
6%.     Also,  find  the  annual  interest  at  4%. 

10.  Find  the  present  worth  of  $6450,  due  in  6  mo.,  with- 
out grace,  money  being  worth  6%. 

11.  Find  the  proceeds  of  a  note  for  $2500,  payable  in  90 
da.,  without  grace,  discount,  5J%. 

12.  How  much  greater  is  the  interest  on  $25,000  for  3  yr. 
6  mo.,  at  6%,  at  compound  interest,  than  at  annual  interest? 

13.  What  was  due  Jan.  1, 1898,  on  a  note  for  $1150,  dated 
Sept.  1,1894,  at  7%  ? 

14.  The  interest  on  a  note  from  Aug.  3  to  Dec.  27,  at  10% 
per  annum,  was  $33.00.     What  was  the  face  of  the  note  ? 

(15.) 

$850T<LV 

PROVIDENCE,  R.  I.,  April  29,  1890. 

For  value  received,  we  promise  to  pay  to 
Webster,  Arnold  &  Co.,  Eight  Hundred  and  Fifty  Dollars, 
ninety  days  after  date,  with  interest  at  6%. 

CHARLES  HATHAWAY. 
JOHN  TODD. 


EXCHANGE  281 

Endorsements:  Oct.  13,  1890,  $40;  Jan.  9,  1891,  $32; 
Aug.  21,  1891,  $125;  Dec.  1,  1891,  $10;  March  16,  1892, 
$80.  What  was  due  Nov.  1 1 ,  1892  ? 

16.  What  is  the  difference  between  the  simple  and  the  com- 
pound interest  of  $2362.75  for  2  yr.  2  mo.  2  da.  at  10%  ? 

17.  Having  sold  15%  of  my  stock  one  mouth,  10%  of  it 
the  next  month,  and  25%  the  third  month,  I  had  remaining 
$2650  worth  of  goods.     What  stock  had  I  before  I  began  to 
sell? 

18.  Discounted  a  note  of  $309.59  for  90  days,  at  10%,  and 
invested  the  proceeds  in  flour  at  $10  per  barrel.     How  many 
barrels  did  I  purchase? 

19.  If  at  7^%  discount  $75.15  is  received  on  a  60-day 
note  20  days  after  its  date,  what  is  the  face? 

20.  What  principal  at  6f  %  interest  will  gain  $85.60  from 
May  4,  1897,  to  Jan.  6,  1898  ? 

21.  If  -jSg-  of  the  price  received  for  an  article  equals  the 
loss,  what  is  the  loss  per  cent.  ? 

22.  Find  the  exact  interest  of  $1200  at  5%  from  Aug.  19, 
1896,  to  March  4,  1898. 

23.  Write  a  note  that  will  be  negotiable  either  in  Pennsyl- 
vania or  New  Jersey. 

EXCHANGE. 

1.  A  Bank  Draft,  or  Bill  of  Exchange,  is  a  written  order 
directing  one  person  to  pay  a  specified  sum  to  another. 

2.  A  Sight  Draft  directs  payment  to  be  made  at  sight  or 
on  presentation. 

3.  A  Time  Draft  directs  payment  to  be  made  at  a  certain 
time  after  sight  or  date. 

4.  An  Acceptance  is  a  draft  with  the  .word  "accepted" 
written  across  its  face,  together  with  the  name  of  the  acceptor, 
who  thus  makes  himself  responsible  for  payment. 


282  PRACTICAL  ARITHMETIC 

5.  The  method  of  paying  by  draft  money  due  at  a  distance 
is  called  Exchange. 

6.  Drafts  are  bought  and  sold,  and  are  described  as  at  par, 
i.e.,  as  having  their  face  value ;  as  at  a  premium,  i.e.,  as  having 
more  than  their  face  Value ;  and  as  at  a  discount,  i.e.,  as  having 
less  than  their  face  value. 

General  Form  of  a  Draft. 


•*  •» 

<Jsi£&'6est-c<^<s>4j&^&ez-s  <^sa-.  ,  May  10,  -/<S  99. 
At  sight  [or  days  after  sight] 

ffl         V 

f  j4  Jernndpr  Richardson        

"^                          «**«<•   07 

Two  hundred       ^^._ 

<z**«z:   c^ia-'ta-e-   •ifo-    «zo€K2- 

100  &*&"* 

***a^  a^ 

100^. 

Simon  Osgood. 

t-Xo-  Brown,  Jones  $•  Co. 

Demand  Draft. 


AMERICAN     I  V<  II  \\4.i:    BANK. 

.,  Sept.  7,    S898. 


Mechanics'  National  Bank, 


S    A    Battaile 

Cashier. 


The  method  of  using  the  above  draft  is  as  follows :  Assume  that  Jona- 
than Wills,  of  St.  Louis,  Mo.,  owing  Wm.  E.  Smith,  of  Philadelphia, 
$274.17,  and  purposing  to  pay  the  debt,  enters  the  American  Exchange 


DOMESTIC  EXCHANGE  283 

Bank  in  St.  Louis,  and,  by  depositing  the  requisite  sum  of  money,  obtains 
the  draft.  He  then  writes  on  the  back  thereof,  "  Pay  to  the  order  of  Wm. 
E.  Smith,"  and  signs  his  own  name.  He  finally  forwards  the  draft  to 
Smith  in  Philadelphia,  who,  taking  it  to  the  Mechanics'  National  Bank 
and  writing  his  name  on  the  back,  receives  the  money. 

Collection  Draft. 


XfjOOX                                                                            ©0VVCCKIO,    <1 

0tt..  March  15,  18  55. 

0 

At  ten  days1  sight,  <\ 

^tfu  to  tft1©  oltie^  o| 

John   Hill 

Two  hundred                           

M,  uj.j^.0,0,  *f)oW'Q.;t6- 

'uatu/e  A,eoevucd,  d'wd  c£mVoe  to  CKKJOU-W 

t  of 

^fo  Henry  Smith, 

New  York,  N.  Y. 

Jo/in  Jm 

The  method  of  using  the  above  draft  is  as  follows :  Assume  that  Henry 
Smith,  of  New  York,  owes  John  Hill,  of  Chicago,  $200,  and  that  Hill,  de- 
siring to  collect  the  money,  prepares  the  above  draft,  and  having  endorsed 
it  thus:  "  Pay  to  George  Jones,  Cashier,  or  order,  for  collection,"  sends  it 
to  the  bank  in  New  York  of  which  Jones  is  cashier.  Jones,  receiving  the 
draft,  presents  it  to  Smith,  who  writes  upon  its  face,  "Accepted,"  signs 
his  name,  and  the  draft  is  said  to  be  honored.  Smith  is  now  under  obliga- 
tion to  pay  to  the  bank  the  $200  at  the  end  of  the  ten  days.  Should  Smith 
fail  to  pay,  the  draft  is  said  to  be  dishonored,  and  is,  in  consequence,  pro- 
tested unless  marked  "  without  protest." 

DOMESTIC    EXCHANGE. 

PROBLEMS. 

1.  What  will  be  the  cost  of  a  sight  draft  on  New  York 
for  $6000  at  f%  premium? 

Process. 

$6000  X  .OOf  =  $22.50,  premium. 

$6000  +  $22.50  =  $6022.50,  cost  of  draft. 


284  PRACTICAL   ARITHMETIC 

2.  What  is  the  cost  of  a  draft  on  Chicago  for  $4200  at 

f %  discount? 

Process. 

|4200  X  .00}  =  $31.50,  discount. 
$4200  —  $31.50  =  $4168.50,  cost. 

3.  Find  the  cost  of  a  draft  for  $1000,  payable  in  60  days 
after  sight,  when  exchange  is  \%  premium,  and  interest  6%. 

Process. 

Draft  ==  $1000.00 

Discount  for  60  da.,  at  6%  =  _     10.00 

$990.00 

Premium  at  J  %  =          2.50 
Cost  of  draft  =    $992.50 

4.  The  draft  was  for  $580 ;  the  time  30  da.  after  sight ; 
the  exchange  is  at  a  premium  of  3%.     Find  the  cost. 

5.  What  will  be  the  cost  in  Philadelphia  of  a  draft  on 
Boston  for  $1800,  payable  60  days  after  sight,  exchange  being 
at  a  premium  of  2%  ? 

6.  What  must  be  paid  in  Detroit  for  a  draft  of  $3000  on 
Boston  at  30  days,  exchange  being  \%  premium? 

7.  If  exchange  on  Chicago  is  \\%  premium,  what  will  be 
the  cost  in  Savannah,  Ga.,  of  a  sight  draft  for  $3000? 

8.  What  will  be  the  cost  of  a  sight  draft  on  New  York 
for  $6400,  at  \\%  premium? 

9.  Find  the  cost  of  a  draft  on  Omaha  for  $1400,  payable 
in  60  days,  when  exchange  is  J%  premium,  and  interest  5%  ? 

10.  Find  the  cost  of  a   draft  on  Baltimore  for  $1237.50, 
payable  in  30  da.  after  sight,  exchange  being  J%  discount, 
and  interest  5%  ? 

11.  A  San  Francisco  merchant  bought  goods  in  New  York 
valued  at  $5284.     What  will  be  the  cost  of  a  3  mo.  draft  for 
the  amount  on  New  York  at  f-%  premium? 


DOMESTIC  EXCHANGE  285 

1 2.  What  must  be  paid  for  a  draft  of  $900  on  New  Orleans 
at  90  da.,  exchange  at  f  %  discount,  and  interest  5%  ? 

13.  If  a  Boston   firm  owes  a  bill  in  Chicago  of  $8750, 
what  must  they  pay  for  a  draft  on  Chicago,  exchange  at  f  % 
premium  ? 

14.  What  must  be  the  face  of  a  draft  to  pay  $500,  exchange 
being  at  1J%  premium? 

Process. 

$1.00  +  .015  =  1.015,  cost  of  a  draft  for  $1.00. 
Hence  $500  -5-  $1.015  =  $492.61  =  the  draft  that  $500 
will  buy. 

15.  Find  the  face  of  a  draft,  drawn  at  30  da.,  that  will  pay 
$369.72,  exchange  being  at  3J%  discount. 

Analysis. 

Discount  of  $1.00  for  30  da.  at  6%  =  .005.  $1.00  —  .005 
=  .995.  .995  —  .0325  =  .9625,  face  that  will  pay  $1.00. 
$369.72  -s-  .9625  =  $384.125,  face  that  will  pay  $369.72. 

16.  How  large  a  sight  draft  can  be  purchased  on  Chicago 
for  $6836  when  the  rate  of  exchange  is  J%  premium? 

17.  What  is  the  face  of  a  90-day  draft  on  Philadelphia 
bought  for  $4600  at  6%,  exchange  \\%  premium? 

18.  What  is  the  face  of  a  draft  on  St.  Paul  for  60  days 
which  may  be  bought  for  $2000,  exchange  being  |%  discount 
and  interest  7%  ? 

19.  How  large  a  draft  on  sight  on  San  Francisco  can  be 
purchased  for  $3500  if  exchange  is  at  J%  premium? 

20.  If  I  pay  $325.05  for  a  draft  payable  60  days  after 
sight,  what  is  the  face  of  the  draft  if  exchange  is  1  %  discount 
and  interest  6  %  ? 

21.  A  draft  payable  90  days  after  sight  was  bought  for 


286  PRACTICAL  ARITHMETIC 

$2756   when  exchange  was  }%  discount  and  interest  6%. 
What  was  its  face? 

22.  If  exchange  is  \%  premium,  how  large  a  draft  will 
$1201.50  buy? 

23.  If  exchange  is  at  If  %  premium,  what  bill  of  exchange 
can  be  bought  for  $762,  in  current  funds,  supposing  a  discount 
of  \%  is  charged  on  the  funds? 


RATIO  AND   PROPORTION. 

1.  Ratio  is  the  relation  which  one  quantity  has  to  another 
of  the  same  kind,  and  is  expressed  by  a  common  fraction,  as 
-|  and  4^.     These  fractions  express  the  ratio  of  2  to  3  and 
12  to  7.     The  same  ratios  may  be  expressed  thus  :  2  :  3  and 
12  :  7.     2  and  12  are  called  antecedents.     3  and  7  are  called 
consequents. 

2.  Since  ratios  have  a  fractional  form,  they  are  governed 
by  the  principles  that  govern  fractions.     -|  =  -f- ;   therefore 
2  :  3  and  6  :  9  express  the  same  ratio. 

3.  Ratio  cannot  exist  between  two  quantities  of  different 
kinds.     If  it  be  required  to  find  the  ratio  between  5  pounds 
and  25  ounces,  the  pounds  must  first  be  reduced  to  ounces,  or 
the  ounces  to  pounds.     5  Ibs.  =  80  oz.     The  ratio  of  80  oz. 
to  25  oz.  =  |f  -  V-. 

4.  A  proportion  is  an  equation  composed  of  two  ratios. 
Since  f  =  y,  the  expression  is  called  a  proportion,  and  the 
terms  8,  4,  18,  and  9  are  called  proportionals.     The  propor- 
tion may  also  be  written  thus :  8:4  =  18:9;  or,  8:4:: 
18:9,  and  is  thus  read  :  "  The  ratio  of  8  to  4  equals  the  ratio 
of  18  to  9  "  ;  or,  "  8  is  to  4  as  18  is  to  9." 

5.  The   first   and   last   terms  of  a  proportion  are  called 
extremes  ;  the  second  and  third  terms  are  called  means. 


EATIO  AND  PROPOKTION  287 

6.  The  proportion  8  :  4  —  18  :  9  is  also  f  =  *£-.     Reduc- 
ing these  fractions  to  a  common  denominator,  we  have  |-^-|  = 

is  x  4.     Since  the  fractions  are  equal  and  the  denominators 
9x4  ^ 

are  equal,  the  numerators  are  equal;  that  is,  8  X  9  =  18  X 
4.  But  8  and  9  are  the  extremes,  and  18  and  4  are  the 
means.  Hence  we  have  the  following  fundamental  principle : 
The  product  of  the  extremes  equals  the  product  of  the 
means. 

7.  No  four  terms  are  proportional  unless,  when  arranged 
as  means  and  extremes,  they  conform  to  the  foregoing  prin- 
ciple.    2,  5,  6,  and  12  are  not  proportionals,  since  they  can- 
not be  arranged  to  make  the  product  of  the  extremes  equal 
the  product  of  the  means. 

8.  If  any  one  of  the  terms  of  a  proportion  is  wanting,  it 
may  readily  be  found.     For  convenience  we  will  call  the  un- 
known term  x,  and  find  what  number  x  equals. 

(a.)  If  6,  8,  and  9  are  the  first,  second,  and  third  terms  of  a 
proportion,  what  is  the  fourth  term?  By  using  x  we  have 
6  :  8  =  9  :  x.  The  product  of  the  extremes  being  equal  to 
the  product  of  the  means,  we  have  the  equation,  6  times  x  =  8 
times  9 ;  or,  Qx  =  72.  Since  6  times  x  =  72,  once  x  =  lf- 
=  12.  Hence  12  is  the  fourth  term,  and  the  completed  pro- 
portion is  6  :  8  =  9  :  12. 

(6.)  Let  the  second  term  be  wanting,  as  in  6  :  x  =  3  :  7. 
Applying  the  principle  we  have  3  times  x  =  6  times  7 ;  or, 
3x  =  42.  x  =  ^f-  =  14,  the  second  term.  The  proportion 
completed  is  6  :  14  =  3  :  7. 

Hence  the  formulae : 

Product  of  Means 


(A.)  Required  Extreme  = 
(B.)  Required  Mean  = 


Given  Extreme 
Product  of  Extremes 


Given  Mean 

To  find  any  required  term  the  other  three  terms  must  be  given ;  hence 
arose  the  old  name,  Rule  of  Three. 


288  PRACTICAL  ARITHMETIC 

EXERCISES. 
Find  the  value  of  x  in  : 

1.  8:  12  =  16:  a?.  9.  20  :  6  =  12  :  x. 

2.  18  :  10  =  x  :  30.  10.  280  :  16  «  140  :  x. 

3.  16  :  x  =  12  :  24.  1 1.  \  :  5  =  x  :  6. 

4.  x  :  5  =  8  :  20.  12.  f  :  x  =  J  :  10. 

5.  x  :  16  =  20  :  80.  13.  6  :  |  =  10  :  x. 

6.  18  :  x  =  14  :  42.  14.  .20  :  .05  =  11  :  x. 

7.  12  :  10  =  x  :  22.  15.  1.7  :  1.9  =  1.5  :  x. 

8.  16  :  8  =  24  :  x.  16.  2.8  :  3.9  =  .07  :  x. 

The  Principles  of  Proportion  Applied  to  Practical 
Problems. 

1.  Most  practical  problems  under  this  rule  involve  the  use 
of  concrete  (denominate)  numbers,  and  in  each  example  there 
are  only  two  different  kinds  of  quantities. 

2.  Every  problem  furnishes  two  like  quantities  and  a  third 
quantity  of  like  denomination  with  the  required  answer. 

ILLUSTRATIONS. 
1.  If  8  Ib.  of  sugar  cost  40  cents,  what  will  20  Ib.  cost? 

Process. 

(a.)  The  ratio  of  the  pounds  equals  the  ratio  of  the  costs. 
(b.)  The  ratio  of  the  pounds  is  8  :  20 ;  hence  the  ratio  of 
costs  is  40  :  x,  since  20  Ib.  cost  more  than  8  Ib.     Therefore 
we  have : 

8  :  20  =  40  :  x.     Or,  40  :  x  =  8  :  20.  Analysis. 

8*  =  800.  If»£=  *>«*" 

1  Ib.  =      5  cts.  ; 
x  =  100,  the  cost  of  20.  20  Ib.  =  100  cts. 

NOTE. — The  pupil  will  observe  that  because  x  is  to  be  greater  than  its 
antecedent,  the  second  term  must  be  greater  than  its  antecedent.  It  is  only 
in  this  way  that  the  equality  of  ratios  can  be  preserved. 


RATIO  AND  PROPORTION  289 

2.  If  6  men  can  do  a  piece  of  work  in  5  days,  in  how  many 
days  can  10  men  do  the  work  ? 

Process. 

We  find  that  the  answer,  or  x,  will  be  less  than  the  third 
quantity,  for  10  men  will  not  require  so  long  a  time  as  6  men. 
We  have,  therefore : 

10:6-6:*.     Or,  6  :  10  =  *  :  5.  Analy8iS' 

If  6  men  =    5  days, 

**•  1  man  =  30  days  ; 

X  =  3  da.  10  men  =    3  days. 

THE  RULE  OP  THREE. 

1.  Let  x  represent  the  required  term  or  answer. 

2.  With  x  and  the  quantity  of  like  denomination  form 
a  ratio. 

3.  Compare  the  two  terms  of  the  ratio,  and  determine 
from  the  conditions  of  the  problem  whether  x  is  greater 
or  less  than  the  other  term. 

4.  "With  the  two  given  like  quantities  form  a  ratio  equal 
to  the  first. 

5.  Express  the  equality  of  the  ratios,  and  apply  formula 
A  or  B,  as  the  case  may  require. 

PROBLEMS. 

NOTE. — Solve  the  following  problems  both  by  analysis  and  proportion. 
Suggestion  :  Let  x  represent  the  fourth  term. 

1.  If  31  yd.  of  cloth  cost  $62,  what  will  21  yd.  cost? 

2.  How  long  will  it  take  24  men  to  do  a  piece  of  work 
that  8  men  can  do  in  12  days  ? 

3.  How  far  can  a  certain  load  be  carried  for  $34,  if  $64 
will  carry  it  100  miles? 

4.  If  231  men  have  provisions  for  8  mo.,  how  long  will 
the  same  provisions  last  308  men  ? 

5.  If  95  cents  will  buy  one  bushel  of  wheat,  how  many 
bushels  will  $11.75  buy? 

19 


290  PRACTICAL  ARITHMETIC 

6.  A  man  owes  $2500,  and  can  pay  only  $1000.     How 
much  does  he  pay  on  a  dollar  ? 

7.  If  28  yd.  of  oil-cloth,  .875  yd.  wide,  cover  a  certain 
floor,  how  many  yards  1.25  yd.  wide  will  cover  the  same  floor  ? 

8.  If  22  bu.  3  pk.  of  corn  be  produced  on  one  acre,  how 
many  acres  will  produce  546  bu.  ? 

9.  If  two  men  earn  $72  in  6  da.,  how  much  will  30  men 
earn  in  the  same  time  ? 

10.  If  12J  tons  of  hay  cost  $180.25,  what  will  81J  tons 
cost? 

Suggestion :  Let  x  represent  the  third  term  in  the  following  problems. 

11.  A  regiment  of  960  men  has  provisions  for  40  days. 
How  long  will  it  last  if  the  regiment  is  reinforced  by  240 
men? 

12.  A  field  can  be  mowed  in  4  days  of  11  hours  each  ;  how 
many  days  of  9  hours  each  will  it  take  ? 

13.  At  the  time  when  a  man  5  ft.  9  in.  in  height  casts  a 
shadow  4  ft.  6  in.  long,  what  is  the  height  of  a  tree  that  casts 
a  shadow  52  ft.  6  in.  long  ? 

14.  If  a  locomotive  runs  96f  miles  in  3J  hours,  how  many 
miles  will  it  run  in  5J  hours? 

15.  A  wheel  makes  75  revolutions  in  5  min.     How  many 
does  it  make  in  an  hour  ? 

16.  A.  can  do  a  piece  of  work  in  6  days,  B.  can  do  it  in  7 
days.     If  B.'s  wages  are  $2.10  per  day,  how  much  should  A. 
receive  per  day  ? 

17.  If  a  5-cent  loaf  of  bread  weighs  8  ounces  when  flour  is 
worth  $5,  what  should  such  a  loaf  weigh  when  flour  is  at  $6  ? 

18.  $  yd.  cost  $£.     Find  the  cost  of  f  yd. 

19.  If  a  cistern  containing  3000  gal.  leak  1  gal.  2  qt.  a 
min.,  how  long  will  it  take  to  empty  it? 

20.  If  42  yd.  of  carpet  2  ft.  3  in.  wide  are  required  for  a 
room,  how  many  yd.  of  carpet  2  ft,  4  in,  wide  will  be  required  ? 


RATIO   AND    PROPORTION  291 

Suggestion  :  Let  x  represent  the  second  term  in  the  following  problems. 


21.  If  a  train,  at  the  rate  of  ^  of  a  mile  per  min.,  requires 
3J  hr.  to  make  a  certain  distance,  how  long  will  it  require  at 
the  rate  of  -fa  of  a  mile  a  min.  ? 

22.  If  a  train  travels  £  of  a  mile  in  18  sec.,  how  many 
miles  an  hour  does  it  travel? 

23.  A.  gains  4  yd.  on  B.  in  running  30  yd.     How  many 
yd.  will  he  gain  while  B.  is  running  97J  yd.  ? 

24.  If  a  man  spends  $276  in  the  three  summer  months, 
how  much  will  he  spend  in  a  year  at  the  same  rate  per  day  ? 

25.  If  28  men  mow  a  field  of  grass  in  12  days,  how  many 
men  will  be  required  to  mow  it  in  8  days  ? 

26.  If  17  men  can  mow  a  field  in  9  days,  how  long  would 
it  take  to  mow  half  of  it  if  5  men  refuse  to  labor  ? 

27.  If  14J  yd.  of  cloth  cost  $19  J,  how  much  will  19|  yd. 
cost? 

28.  If  ^  of  a  ship  costs  £273  2s.  6d.,  what  will  -fo  of  her 
cost? 

29.  If  2J  gal.  of  molasses  cost  65  cents,  what  will  3J  hhd. 
cost? 

30.  If  a  steeple  150  ft.  high  casts  a  shadow  210  ft.,  what 
is  the  length  of  the  shadow  cast,  at  the  same  time,  by  a  staff 
5ft.  high? 

Suggestion  :  Let  x  represent  the  first  term  in  the  following  problems. 

31.  If  the  interest  of  $600  for  6  mo.  is  $15,  what  principal 
will  gain  $64  in  the  same  time  ? 

32.  If  15  J  yd.  of  silk  that  is  f  yd.  wide  will  make  a  dress, 
how  many  yards  of  muslin  that  is  1  J  yd.  wide  will  be  required 
to  line  it? 

33.  If  I  borrow  $500  and  keep  it  1  yr.  4  mo.,  for  how  long 
a  time  should  I  lend  $240  as  an  equivalent  for  the  favor  ? 


292  PEACTICAL  ARITHMETIC 

34.  A  butcher  in  selling  meat  sells  14^  oz.  for  a  pound. 
How  much  does  he  cheat  a  customer  who  buys  of  him  to  the 
amount  of  $30  ? 

35.  In  what  time  can  a  man  pump  64  hhd.  of  water  if  he 
can  pump  12  hhd.  in  2  hr.  15  min.  ? 

36.  How  many  men  can  do  in  24  days  a  piece  of  work 
which  would  employ  40  men  6  days? 

37.  If  J  of  f  of  6J  bbl.  of  beef  cost  $78,  how  much  will 
|  of  |  of  3|  bbl.  cost? 

38.  If  450  tiles,  each  1 2  in.  square,  will  pave  a  cellar,  how 
many  tiles  that  are  9  in.  by  8  in.  will  pave  the  same  ? 

39.  If  a  distance  of  48  miles  is  represented  on  a  map  by 
If  in.,  what  distance  is  represented  on  the  same  map  by  7  J  in.  ? 

40.  Twenty-four  men  in  30  days  can  finish  a  piece  of  work. 
After  16  days  11  men  quit  work.    In  how  many  days  can  the 
rest  finish  the  work  ? 

COMPOUND   PROPORTION. 

A  Compound  ratio  indicates  the  product  of  two  or  more 
simple  ratios  ;  for  instance,  -f-  X  -f  is  a  compound  ratio,  being 
the  product  of  the  simple  ratios  ^  and  -J,  written  \\\- 

A  Compound  proportion  has  one  of  its  ratios  compound. 

ILLUSTRATION. 

If  5  men  build  a  wall  6  ft.  high  in  7  days  of  8  hr.,  in  how 
many  days  of  9  hr.  can  10  men  build  a  wall  11  ft.  high? 

Process. 

The  second  ratio  is  simply  7  da.  :  x  da. 
The  first  ratio  is  compound,  and  we  construct  it  as  follows : 
We  write  10  :    5,  for  10  men  require  less  time  than  5  men. 
We  write    6  :  11,  for  11  ft.  require  more  time  than  6  ft. 
We  write    9  :    8,  for  9  hr.  per  day  require  fewer  days  than 
8  hr. 


COMPOUND  PROPORTION  293 

(10:    5^ 

Hence  the  proportion  is :  -[     6  :  11  V  :  :  7  :  x. 

(    9:    8J 
2 

By  formula  A,  a?  =  *  ™"  g^*  = 


?      3 

r>      A       i     •        7  X  5  X  11  X  8         Rlq    , 
By  Analysis  :  ^—-^rr      T^  —  oM  da. 


Since  5  men  require  7  da.,  10  men  require  a  shorter  time,  i.  e.,  T5<y  of  7 
da. ;  since  6  ft  ,  etc. 

PROBLEMS. 

1.  If  6   men  can   mow  24  acres  of  grass  in  2  days,  by 
working  10  hours  per  day,  how  many  days  will  it  take  7  men 
to  mow  56  acres  by  working  12  hrs.  per  day? 

2.  If  4  men  mow  15  A.  in  5  da.  of  14  hr.,  in  how  many 
da.  of  13  hr.  can  7  men  mow  19  J  A.  ? 

3.  If  810  bricks,  8  in.  long  and  4  in.  wide,  are  required 
for  a  walk  36  ft.  long  and  5  ft.  wide,  how  many  bricks  will 
be  required  for  a  walk  66  ft.  long  and  4  ft.  \vide? 

4.  If  the  interest  on  $640  for  4  yr.  6  mo.  is  $172.8,  what 
is  the  interest  on  $820  for  2  yr.  8  mo.  at  the  same  rate  ? 

5.  If  it  requires  275  yd.  of  cloth  f  yd.  wide  to  make  75 
garments,  how  many  yards  of  cloth  1J  yd.  wide  will  it  require 
to  make  215  such  garments? 

6.  If  it  costs  $2.40  to  carry  20  cwt.  50  miles,  what  will 
it  cost  to  carry  40  cwt.  40  miles  at  the  same  rate  ? 

7.  If  1 2  candles,  8  weighing  a  pound,  last  from  5  o'clock 
to  11,  how  many  candles,  6  weighing  a  pound,  will  last  from 
7  o'clock  to  11? 

8.  A  farmer  owning  25  horses  traded  them  for  sheep.     If 
3  horses  are  worth  1 2  cows,  6  cows  are  worth  42  pigs,  and  25 


294  PRACTICAL  ARITHMETIC 

pigs  are  worth  30  sheep,  how  many  sheep  did  he  get  for  his 
horses  ? 

9.  If  200  men  in  12  days  of  8  hr.  each  can  dig  a  trench 
160  yd.  long,  6  yd.  wide,  and  4  yd.  deep,  in  how  many  days 
of  10  hr.  will  90  men  dig  a  trench  450  yd.  long,  4  yd.  wide, 
and  3  yd.  deep? 

10.  5  compositors,  in  16  da.  of  14  hr.  each,  can  compose 
20  sheets  of  24  pages  in  each  sheet,  50  lines  in  a  page,  40 
letters  in  a  line.     In  how  many  days  of  7  hr.  each  will  10 
compositors  compose  a  volume  containing  40  sheets,  16  pages 
in  a  sheet,  60  lines  in  a  page,  50  letters  in  a  line  ? 

11.  At  6%,  what  principal  will  gain  $27  in  9  months? 

12.  If  12  horses  eat  10  bu.  of  oats  in  8  da.,  how  many 
bushels  will  30  horses  eat  in  40  days  ? 

13.  If  it  takes  22  reams  of  paper  to  make  1000  copies  of 
a  book  of  11  sheets,  how  many  reams  will  be  required  to 
make  4500  copies  of  a  book  of  7  sheets  ? 

14.  If  a  field  60  rods  long  and  20  rods  wide  cost  $500, 
what  will  a  field  15  chains  long  and  8  chains  wide  cost? 

15.  If  a  piece  of  iron  7  ft.  long,  4  in.  wide,  and  6  in.  thick 
weighs  600  lb.,  how  much  will  a  piece  of  iron  weigh  that  is 
16  ft.  long,  8  in.  wide,  and  4  in.  thick? 

16.  If  a  6-cent  loaf  weighs  8  ounces  when  wheat  is  $1.25 
per  bu.,  how  much  bread  may  be  bought  for  50  cents  when 
wheat  is  $1.00  per  bushel? 

17.  A  ship's  crew  of  32  men,  at  a  daily  allowance  of  3  lb. 
to  each  man,  have  provisions  enough  for  45  days.     If  they 
now  rescue  a  crew  of  16  men,  what  can  be  allowed  each  man 
daily  to  make  the  provisions  hold  out  40  days  ? 

18.  4000  copies  of  a  book,  containing  420  pages,  were 
printed  from  650  reams  of  paper ;  how  many  reams  of  paper 
would  have  been  required  to  print  7000  copies,  containing 
528  pages,  of  the  same  size? 


CAUSE  AND  EFFECT  295 


19.  If  $500  will  gain  $16.50  in  4  mo.  12  da.,  at  9%,  how 
much  will  $750  gain  in  2  yr.  9  mo.  8  da.,  at  6%  ? 

20.  If  3280  42-lb.  shot  cost  $3000,  how  many  32-lb.  shot 
can  be  bought  for  $4200? 

21.  How  many  hours  a  day  must  5  men  work  to  mow  a 
field  in  8  days,  that  7  men  can  mow  in  6  days  of  10  hours? 

22.  If  25  horses  can  consume  a  bin  of  grain  in  40  days,  in 
what  time  will  a  bin  of  twice  the  size  be  consumed,  if  7  horses 
are  added  when  the  grain  is  f  eaten  ? 

23.  $600  gains  $72  in  2  years.     In  how  many  years  at  the 
same  rate  will  $92  gain  $54  ? 

CAUSE    AND    EFFECT. 

Since  like  causes  produce  like  effects,  we  have  the  following 
general  formula  : 

1st  cause  :  2d  cause  :  :  1st  effect  :  2d  effect. 

ILLUSTRATIONS. 

1.  If  4  men  earn  $144,  how  much  will  6  men  earn  in  the 
same  time  and  at  the  same  rate  ? 

Process. 

Let  x  be  the  required  effect,  representing  what  6  men  will 
earn,  and  we  have  : 

1st  c.   2d  c.      1st  ef.    2d  ef. 
4    :    6  =  144    :   x; 
4x  =  864; 
x  =  216,  ans. 

2.  If  4  men  earn  $144  in  12  days,  how  much  will  6  men 
earn  in  10  days  at  the  same  rate  ? 

NOTE.  —  Here  there  are  compound  causes,  consisting  of  men  and  days. 


296  PRACTICAL  ARITHMETIC 

Process. 

Let  x  dollars  be  the  required  effect  of  6  men  and  10  days, 
and  we  have  : 

5         36 

12  •  10  /  ::  144  :  x'       x  =      f  x  ;? —  ==  $^®- 

PROBLEMS. 

1.  If  3  workmen  can  board  4  weeks  for  $54,  how  many 
can  board  13  weeks  for  $585  ? 

{c.     c.  -\       ef.      ef. 
3  :  x    \  ::  54  :  585. 
4:  13  J 

2.  If  36  men  earn  $1296  in  13  days,  how  much  will  42 
men  earn  in  87  days  ? 

3.  If  12  horses  consume  40  bu.  of  oats  in  8  days,  how 
long  will  140  bu.  of  oats  last  16  horses? 

Suggestion  :  Let  x  days  be  a  cause. 

4.  If  it  cost  $15  to  carry  20  tons  1 J  miles,  what  will  it  cost 
to  carry  400  tons  J  of  a  mile  ? 

5.  If  A.  can  do  f  of  a  piece  of  work  in  5  da.,  working  8 
hr.  a  da.,  how  long  will  it  take  him  to  do  the  whole  piece, 
working  10  hr.  a  day  ? 

6.  If  12  horses  in  5  da.  draw  44  loads  of  stone,  how  many 
horses  will  draw  132  loads  the  same  distance  in  18  da.? 

NOTE. — If  additional  practice  is  needed  in  applying  the  principle  of 
cause  and  effect,  any  of  the  previous  problems  in  proportion  may  be  used. 

PROPORTIONAL  PARTS. 

A  number  may  be  divided  into  parts  which  are  proportional 
to  two  or  more  given  numbers. 

ILLUSTRATIONS. 

1.  Divide  the  number  180  into  three  parts  that  shall  be  to 
one  another  as  3,  4,  and  5. 


PROPORTIONAL  PARTS  297 

Process. 

Let  180  =  3  -j-  4  +  5  =  12  parts. 
If  12  parts  =  180. 

3  parts  =  T32-  of  180  =  45. 

4  parts  =  -fa  of  180  =  60. 

5  parts  =  -fj  of  180  =  75. 

180. 

2.  Divide  940  in  the  proportion  of  ^,  -J-,  J. 

Process. 
The  L.  C.  D.  of  the  fractions  is  60.     £  =  ££ ;  £  =  f# ; 

940  =  12  +  20  +  15  =  47  parts. 
If  47  parts  =  940. 

12  parts  =  if  of  940  =  240. 
20  parts  =  fj  of  940  —  400. 
15  parts  =  ^  of  940  =  300. 

940. 

PROBLEMS. 

1.  Divide  60  into  two  parts  that  are  to  each  other  as  5 
and  7. 

2.  Divide  1200  into  parts  proportional  to  11,  12,  13,  14. 

3.  Divide  780  into  parts  proportional  to  J,  J,  J. 

4.  Three  men  caught  120  fish.     How  many  did  each  catch, 
their  proportions  being  as  2,  1  J,  and  J  ? 

5.  Divide  a  profit  of  $13,384  among  three  partners,  the  first 
owning  -^,  the  second  -|-|,  the  third  -|-|. 

6.  Divide  $9  into  parts  that  are  to  each  other  as  .05,  .10, 
.25,  and  .50. 

7.  A  house,  a  farm,  and  a  store  cost  $18,000.     The  farm 
cost  twice  as  much  as  the  house,  and  the  store  three  times  as 
much  as  the  house.     How  much  did  each  cost  ? 


298  PRACTICAL  ARITHMETIC 

8.  Three  men  agree  to  pay  $60  rent  for  a  pasture  lot ;  the 
first  pastures  3  cows,  the  second  5  cows,  and  the  third  4  cows. 
How  much  should  each  pay  ? 

9.  If  gunpowder  contains  nitre,  charcoal,  and  sulphur  in 
the  proportion  of  15,  3,  and  2,  and  if  in  a  quantity  of  gun- 
powder there  is  20  cwt.  of  charcoal,  find  the  weight  of  nitre 
and  sulphur  therein. 


PARTNERSHIP. 

1.  Partnership  is  of  two  kinds, — Simple  and  Compound. 

2.  It  is  simple  when  the  capital  of  the  partners  continues 
in  the  business  for  the  same  time. 

3.  It  is  compound  when  the  capital  of  the  partners  con- 
tinues in   the  business  for  different  lengtlis  of  time.      Time, 
therefore,  has  to  be  considered  in  the  proportional  division  of 

gains  or  losses. 

ILLUSTRATIONS. 

1.  Two  men,  A.  and  B.,  enter  into  partnership,  and  gain  in 
1  yr.  $500.     What  part  of  the  gain  did  each  own  if  A.'s 
capital  was  $3000  and  B.'s  $2000? 

Process. 

A.'s  capital  is  to  B.'s  as  3  to  2. 
$500  =  3  +  2  =  5  equal  parts. 
If  5  parts  =  $500, 

3  parts  =  |  of  $500  =  $300,  A.'s  share; 

2  parts  =  -f  of  $500  =  $200,  B.'s  share. 

2.  But  suppose  A.  had  $3000  in  business  for  3  yr.,  and  B. 
had  $2000  in  business  for  2  yr. ;  how  would  $1300  gain  be 

divided  proportionally? 

Difference  of  time  must  be  considered  as  well  as  difference  of  capital. 


I 

PARTNERSHIP  299 

Process. 

$3000  for  3  yr.  =  $9000  for  1  yr. 
$2000  for  2  yr.  =  $4000  for  1  yr. 

$9000  :  $4000  =  9:4. 
$1300  =  9  -|-  4  =  13  equal  parts. 
If  13  equal  parts  =  $1300, 

9  parts  =  ^  of  $1300  =  $900,  A.'s. 
4  parts  =  -^  of  $1300  =  $400,  B.'s. 

PROBLEMS. 

1.  A.  and  B.  engage  in  trade.     A.  furnishes  $300,  and  B. 
$400  of  the  capital.     They  gain  $182.     What  is  each  one's 
share  of  the  gain  ? 

2.  Two  persons  form  a  partnership.     A.  puts  in  $450  for 
7  mo.  and  B.  $300  for  9  mo.     They  lose  $156.     How  much 
is  each  man's  share  of  the  loss  ? 

3.  A.,  B.,  and  C.  paid  $220.50  for  a  pasture.     A.  put  in 
9  cows  for  2J  mo.,  B.  12  cows  for  2  mo.,  C.  18  cows  for  1J 
mo.     How  much  of  the  rent  ought  each  to  pay  ? 

4.  C.,  D.,  and  E.  formed  a  partnership  for  carrying  on 
business.     C.  furnished  $800  for  6  mo.,  D.  $1800  for  8  mo., 
and  E.  $1500  for  4  mo.     They  gained  $940.     How  should 
the  gain  be  divided  ? 

5.  A.,  B.,   C.,  and  D.   traded  in  company.      A.  put   in 
$7500,  B.  $7000,  C.  $9500,  and  D.  $8000.     What  was  each 
partner's  share  of  a  profit  amounting  to  $9280  ? 

6.  Three  men  buy  a  house  for  $1200.     A.  furnishes  $600, 
B.  $400,  C.  $200.     They  sell  the  house  for  $1500.     How 
much  money  should  each  receive? 

7.  M.  and  N.  entered  into  partnership.     M.  put  $200  into 
the  business  for  5  mo.  and  N.  $300  for  4  mo.     They  gained 
$132.     Find  the  share  of  each? 


300  PRACTICAL    ARITHMETIC 

8.  Two  men  hire  a  pasture  for  $42.     One  puts  in  twice 
as  many  head  of  cattle  as  the  other.     What  should  each  pay  ? 

9.  A.,  B.,  and  C.  buy  a  house  for  $15,000.     A.  supplies 
$4000,  B.  $5000,  C.  the  remainder.     The  yearly  rental  being 
$1000,  to  what  part  of  it  is  each  entitled? 

10.  E.,  F.,  and  G.  bought  a  block  of  stores  for  $46,000. 
E.  furnished  f  of  the  money,  F.  $11,500,  and  G.  the  rest. 
The  property  was  sold  for  $48,300.     What  was  the  gain  of 
each? 

11.  A.,  B.,  and  C.  engaged  in  manufacturing.     A.  invested 
$4500  for  6  mo.,  B.  $5000  for  8  mo.,  and  C.  $6500  for  7  mo. 
They  gained  $4500.     Find  each  partner's  gain. 

12.  The  profits  were  $4800.     Patterson's  share  was  $3000. 
How  many  eighths  of  the  capital  did  he  own  ? 

13.  X.  and  Y.  hire  a  bicycle  for  $4.50  a  week.     If  X. 
uses  it  on  Tuesday  and  Friday,  and  Y.  the  rest  of  the  week, 
except  Sunday,  what  does  each  pay  ? 

14.  Smith  and  Jones  united  in  a  partnership.     Smith  con- 
tributed $240  for  8  mo.,  and  B.  $560  for  5  mo.     They  lost 
$118.     How  much  did  each  man  lose? 

15.  A  man  dies  owing  three  creditors,  $8050,  $2970,  and 
$7170,  respectively.     If  his  assets,  after  deducting  expenses, 
are  $13,646,  how  much  will  each  creditor  receive? 

16.  A.  and  B.  enter  into  partnership  with  capitals  of  $3500 
and  $8700.     A.  is  to  have  12%  of  the  profits  for  his  services 
as  manager.     Divide  a  gross  profit  of  $1906.25. 

17.  S.  and  T.  engaged  in  speculation.     S.  employed  $1260 
for  8  mo.,  and  T.  $980  for  6  mo.     They  lost  $957.60.     Ap- 
portion this  loss. 

18.  A  man  failing  in  business,  paid  50  cents  on  the  dollar. 
With  assets  of  $40,000,  how  much  would  X.,  Y.,  and   Z. 
receive,  whose  claims  against  him  were  respectively  $2000, 
$3000,  and  $4000? 


PARTNERSHIP  301 

19.  Apportion  a  loss  of  $5600  among  three  partners  whose 
capital  was  invested  in  the  proportion  of  1,  4,  and  5. 

20.  A.  commenced  business  January  1st,  with  a  capital  of 
$3400.     April  1st  he  took  B.  into  partnership  with  a  capital 
of  $2600  ;  at  the  expiration  of  the  year,  they  had  gained  §750. 
What  is  each  one's  share  of  the  gain  ? 

21.  Our  standard  gold  coin  consists  of  900  parts  of  gold, 
90  parts  silver,  10  parts  copper.     What  is  the  quantity  of 
each  metal  in  50  pounds  of  coin  ? 

22.  A.  and  B.,  contractors,  received  $857.50  for  grading  a 
roadway.     A.  furnished  5  men  20  days,  and  6  others  for  15 
days;  B.  furnished  10  men  for  12  days,  and  9  others  for  20 
days.     What  was  each  contractor's  share? 

23.  C.  and  D.  form  a  partnership.     C.  invests  $5000 ;  D. 
$10,000.     During  the  year  C.  draws  $1500  of  the  profits  and 
D.  draws  $1000.     At  the  end  of  the  year  the  business  is  dis- 
posed of  for  $20,000.     What  amount  should  each  receive  ? 

24.  P.,  Q.,  and  R.  buy  a  lot  for  $600.     After  selling  it,  P. 
receives  $220  as  his  share  of  the  proceeds,  Q.  receives  $280 
and  R.  $300.     How  much  did  each  invest  originally? 

25.  A  partnership  is  formed  between  A.,  with  a  capital  of 
$1500,  and  B.,  with  a  capital  of  $4000.     Six  months  there- 
after they  take  in  C.,  with  a  capital  of  $4000.     How  should 
a  profit  of  $3500  be  divided  at  the  end  of  the  year? 

26.  V.  and  W.  rented  a  field  for  a  year  for  $200.     Y.  put 
in  6  horses  for  the  whole  time,  W.  put  in  5  horses  for  11 
months  and  3  horses  for  5  months.     How  much  of  the  rent 
had  each  to  pay  ? 

27.  A.,  B.,  and  C.  entered  into  partnership  for  one  year. 
A.  put  in  $5000,  B.  $6000,  and  C.  $4000.     At  the  end  of  six 
months  A.  withdrew  $2000,  and  C.  put  in  $8000  more.     The 
profits  at  the  end  of  the  year  were  $6000.     What  was  each 
man's  share? 


302  PRACTICAL   ARITHMETIC 

28.  B.  and  C.,  trading  together,  find  their  stock  to  be 
worth  $3500,  of  which  C.  owns  $2100.  They  have  gained 
40%  on  their  first  capital.  What  did  each  put  in? 


AVERAGES. 

An  average  is  expressed  by  the  ratio  of  the  sum  of  two  or 
more  quantities  to  the  number  of  the  quantities. 

ILLUSTRATION. 

A  Fahrenheit  thermometer  registered  30°  at  8  A.M.,  56°  at 
M.,  and  40°  at  6  P.M.  What  was  the  average  temperature  of 
the  day  between  8  and  6  o'clock  ? 

Process. 

1.  The  sum  of  the  quantities  is  30  +  56  +  40  =  126. 

2.  The  number  of  them  is  3. 

3.  Their  average  is  if*  =  42. 
Therefore  the  average  temperature  is  42°. 

PROBLEMS. 

1.  A  tax-collector  received  on  Monday,  $430.74 ;  on  Tues- 
day, $380.88 ;  on  Wednesday,  $448.60 ;  on  Thursday,  $420.79 ; 
on  Friday,  $367.44 ;  on  Saturday,  $508.73.     What  did  he 
receive  daily  on  an  average? 

2.  In  a  school  the  largest  attendance  present  for  each  of  six 
months  was  as  follows:  1st  mo.,  125;  2d  mo.,  130;  3d  mo., 
128;  4th  mo.,  125;  5th  mo.,  132;  6th  mo.,  122.     Find  the 
average  attendance  for  the  6  months. 

3.  A  goldsmith  combined  2  oz.  of  gold  16  carats  fine,  2  oz. 
18  carats  fine,  and  6  oz.  22  carats  fine.     What  is  the  fineness 
of  the  composition  ? 


AVERAGING  OR  EQUATING  OF  PAYMENTS         303 

4.  If  one  dozen  eggs  weigh  1  Ib.  2  oz.,  what  is  their  aver- 
age weight? 

o.  If  a  man  owes  a  debt,  due  in  3  mo.,  and  a  like  debt,  due 
in  5  mo.,  when  may  he  pay  both  debts  at  once  ? 


AVERAGING  OR  EQUATING  OF  PAYMENTS. 

The  averaging  or  equating  of  payments  due  at  different 
times  consists  in  finding  an  equitable  time  for  including  all 
payments  in  one. 

ILLUSTRATION. 

A.  owes  B.  $250,  due  in  3  mo.,  and  $350,  due  in  5  mo. 
Find  the  average  term  of  credit  ? 

NOTE. — If  the  payments  were  equal,  the  average  term  of  credit  would  be 
liJ!  —  4  mo.  Since  they  are  not  equal,  we  must  consider  both  time  and 
payments. 

Short  Process.  Explanatory  Process. 

$250  for  3  mo.  =  1  $1  f     (    750  mo. 

250  X  3  =      750  $350  for  5  mo.  =  /  1 1750  mo. 

350  X  5  =    1750  Total,  $1  for  2500  mo. 

If  the  term  of  credit  on  $1  be  2500  mo., 
the  term  of  credit  on  $600  is  ^  of  2500 
mo.  =  3gffi  =  -2g5-  =  4£  mo. 


Hen  ce  the  rule,  briefly  stated,  is  : 

1.  Multiply  the  debts  by  the  terms  of  credit. 
2.  Divide  the  sum  of  the  products  by  the  sum  of  the 
debts. 

PROBLEMS. 

1.  Equate  the  time  for  payment  of  $400,  due  in  3  mo. ; 
$600,  due  in  7  mo. ;  aud  $300,  due  in  10  mo. 

2.  Find  the  average  time  of  payment  of  $3500,  due  in  5 
mo. ;  of  $1600,  due  in  8  mo. ;  of  $1500,  due  in  10  mo. ;  and 
of  $600,  due  in  9  mo. 


304  PRACTICAL   ARITHMETIC 

3.  Mr.  Jenkins  has  bought  $1200  worth  of  goods  on  P> 
months'  credit  and   $600  worth  on   3  months'  credit.     For 
what   time  should   he   give   a  note   for  the  whole  amount, 
$1800? 

4.  $1680  is  to  be  paid  in  four  equal  instalments,  in  1,  2,  3, 
and  4  mo.  respectively.     Equate  the  time. 

5.  $500  is  due  in  8  mo.,  $900  in  6  mo.,  $1000  in  3  mo., 
$1200  in  cash  [1200  X  0  =  0].     Find  the  term  of  credit  for 
a  single  payment  of  the  whole  indebtedness. 

6.  Equate  the  time  for  the  payment  of  $5000,  due  Feb. 
1  ;  of  $4000,  due  June  1  ;  of  $3000,  due  Aug.   1,  and  of 
$3000,  due  Oct.  1. 

Suggestion  :  Count  time  from  Feb.  1. 

7.  A  person  owes  a  certain  sum,  of  which  J  is  payable  in 
8  mo.,  J  in  9  mo.,  and  the  balance  in  12  mo.     Equate  the 
time  of  payment. 

8.  Johnson  &  Co.  sold  a  bill  of  lumber  on  the  following 
terms:    $1500  cash,  $3000  payable  in  30  days,  and  $2000 
payable  in  90  days.     When  may  the  whole  debt  be  cancelled 
by  one  payment  ? 

9.  If  a  person  lends  me  $250  for  8  mo.,  for  how  long 
ought  I  to  lend  him  $480  as  an  equivalent? 

1 0.  I  bought  on  July  5th  goods  to  the  amount  of  $2400. 
$630  was  to  be  paid  at  once,  $820  in  8  mo.,  and  $950  in  9 
mo.     What  is  the  equated  time  for  the  payment  of  the  whole? 

11.  A  man  owes  $600,  of  which  J  is  to  be  paid  in  1   yr., 
and  the  remainder  in  2  yr.     Equate  the  time,  and  find  the 
present  value,  money  being  worth  6%. 

12.  I  bought  bills  of  goods  as  follows:  June  1,  $250,  on 
3  mo.  credit;  July  5,  $300,  on  3  mo.  credit;  Aug.  6,  $150, 
on  3  mo.  credit ;  Oct.  2.,  $400,  on  2  mo.  credit.     Find  the 
equated  time  of  payment. 


AVERAGING  OR  EQUATING  OF  PAYMENTS         305 

Process.  Explanation. 

250  X     0^00000  1.  Add  the  terms  of  credit  to  their 

300  X  34  =  10200  respective  dates. 

1  50  y  6f>  —  -     9900  ^'  -F1*110^  the  interval  between  the 

~  earliest  resultins  date  and  each  of 


v  Q9 

the  other  dates. 


1100  56900  3.  Multiply  the  debts  by  their  re- 

5g900  -f-  1100  =  51-jSj-  spective  intervals,   and  proceed   as 

Sept.  1  +  51T8r  =  Oct.  23 

13.  Mr.  B.  bought  goods  as  follows  :  April  15,  $150  on 
2  mo.  credit  ;  May  10,  $200  oil  3  mo.  credit  ;  June  5,  $250 
on  4  mo.  credit.     Fiud  the  equated  date  of  payment. 

14.  What  is  the  average  time  at  which  the  following  bills 
become  due  :  Feb.  1,  1898,  $200  on  1  mo.  credit  ;  March  10, 
1898,  $500  on  3  mo.  credit;  April  12,  1898,  $275  on  2  mo. 
credit;  and  May  1,  1898,  $400  on  4  mo.  credit? 

15.  I  owe  Mr.  Wilson  $100,  to  be  paid  on  the  15th  of 
July;  $200  on  the  15th  of  August,  and  $300  on  the  9th  of 
September.     What  is  the  mean  time  of  payment? 

16.  Find  the  equated  time  for  the  payment  of  $112.30 
due  July  6,  $115.25  due  July  30,  $232.15  due  Sept.  4,  and 
$102.36*due  Oct.  1. 

17.  A  merchant  bought  goods  as  follows  :  Mar.  19th,  $350 
on  4  mo.  ;  Apr.  1st,  $430  on  130  da.  ;  May  16th,  $540  on 
95  da.  ;  June  10th,  $730  on  3  mo.  ;  what  is  the  average  time 
for  the  payment  of  the  whole  ? 

18.  $1200  worth  of  mdse.,  bought   Nov.   5,  and  $1000 
worth  bought  on  the  following  Jan.  9,  have  a  credit  of  2  mo. 
When  may  both  be  paid  at  once  ? 

19.  A  man  bought  the  following   bills  of  goods:  Jan.  15, 
$600  on  2  mo.  credit;  Feb.  1,  $300  on  3  mo.  credit;  March 
25,  $550  on  30  da.  credit  ;  and  April  8,  $400  on  60  da.  credit. 
Find  the  equated  time  of  payment. 

20 


306  PRACTICAL  ARITHMETIC 

20.  Find  the  equated  time  of  payment  for  the  following 
obligations : 

1.  $400,  due  June  15  ;  $375,  due  July  11 ;  $195,  due 

Sept.  4. 

2.  $1394.50,  due  Dec.  1, 1898  ;  $129.80,  due  Dec.  10, 

1898 ;  $960,  due  Feb.  1,  1899. 

21.  A.  owes  $600,  due  in  8  mo.     If  he  pays  $160  in  3  mo. 
and  $120  in  6  mo.,  when  should  he  pay  the  balance? 

8  mo.  —  3  mo.  =  5  mo. 
8  mo.  —  6  mo.  =  2  mo. 
Therefore  A.  has  to  his  credit : 

$160  for  5  mo.  =  $1  for  800  mo.  )       fl1  „ 

(»-ion.p     o  ai  f     nAf\          f  —  $1  lor  1040  mo. 

$120  for  2  mo.  =  $1  for  240  mo.  J 

But  A.  still  owes  $600  —  280  =  $320. 
$1  for  1040  mo.  =  $320  for  %4<f  ==  3}  mo.  (after  8  mo.). 

22.  B.  owes  $1600,  due  in  5  mo. ;  $2400  due  in  7  mo.     If 
at  the  end  of  5  mo.  he  pays  $2800,  when  should  the  balance 
be  paid  ? 

23.  A  man  owes  $2000,  due  in  8  mo.     He  pays  $500  in 
2  mo.  and  $800  in  3  mo.     When  in  equity  should  he  pay  the 
balance  ? 

24.  A.  owed  B.  $2000,  payable  in  4  mo.,  but  at  the  end  of 
1  mo.  he  paid  him  $500,  at  the  end  of  2  mo.  $500,  and  at  the 
end  of  3  mo.  $500.     In  how  many  months  is  the  balance  due 
him? 

25.  A.  owes  $800,  due  in  5  mo. ;  $1200,  due  in  7  mo.    If 
at  the  end  of  5  mo.  he  pays  $1400,  when  should  the  balance 
be  paid? 

26.  A  merchant  owes  $5400,  due  in  9  mo.     If  he  pays 
$2300  in  4  mo.,  $2000  in  5  mo.,  and  $600  in  7  mo.,  when 
should  he  pay  the  balance  ? 

27.  A  modiste  bought  goods  to  the  amount  of  $425  on  a 
credit  of  20  da.  and  $380  on  a  credit  of  30  da.     At  the  end 


AVERAGING  OR  EQUATING  OF  PAYMENTS         307 

of  15  da.  she  paid  $450,  and  at  the  end  of  20  da.  she  paid 
$150.     When  can  the  remainder  be  equitably  paid? 

28.  What  is  the  average  date  of  payment  for  the  following 
three  notes:  March  10,  1898,  $240;  April  12,  1898,  $260; 
May  14,  1898,  $320? 

29.  I  bought  goods  to  the  amount  of  $1200  on  the  fol- 
lowing terms :   J  payable  in  cash,  J-  payable  in  2  mo.,  the 
balance  in  6  mo.     When  may  the  whole  in  equity  be  paid  at 
once  ? 

30.  I  owe  $600,  due  in  5  mo. ;  $1000,  due  in  10  mo.,  and 
$1200,  due  in  7-f  mo.     What  is  the  average  term  of  credit? 


INVOLUTION. 

INDUCTIVE  STEPS. 

1.  In  the  equation  3X3  =  9,  there  are  how  many  equal 
factors  ?     What  is  the  product  of  those  factors  ? 

2.  In  the  equation  5  X  5  X  5  =  125,  how  many  times  is 
5  taken  as  a  factor?     What  is  125  called ? 

3.  The  product  of  equal  factors  is  also  called  the  power. 

4.  Find  the  product  or  power  of  6  taken  twice  as  a  factor. 

5.  Find  the  power  of  six  taken  3  times  as  a  factor  ? 

6.  When  a  number  is  taken  twice  as  a  factor,  the  product 
is  called  the  second  power  of  the  number. 

7 .  When  a  number  is  taken  3  times  as  a  factor,  the  product 
is  called  the  third  power  of  the  number ;   when  taken  four 
times,  the  fourth  power,  and  so  on. 

8.  Write  the  second  power  of  2 ;  of  3 ;  of  4 ;  of  5 ;  of 
7  ;  of  8  ;  of  9. 

9.  Write  the  third  power  of  2  ;  of  3  ;  of  4  ;  of  5  ;  of  7  ; 
of  8 ;  of  9. 


308  PRACTICAL    ARITHMETIC 

10.  What  is  the  product  of  f  by  f  ?     Of  f  by  f  by  f  ? 

11.  What,  then,  is  the  second  power  of  f  ?     Third  power? 

12.  What  is  the  second  power  of  -|  ?     Third  power  ? 

13.  The  equation  2x2X2  =  8,  expressing  the  third 
power  of  2,  is  commonly  written  thus  :  23  =  8.     The  3,  indi- 
cating the  number  of  times  2  is  taken  as  a  factor,  is  called  the 
Exponent. 

14.  Write  an  equation  showing  by  an  exponent  the  third 
power  of  4 ;  the  fourth  power  of  5 ;  the  fifth  power  of  6. 

15.  For   the  reason   that  the   product  of  two   equal  fac- 
tors equals  the  area  of  a  square,  and  the  product  of     _s 

s\ — -71     three  equal  factors  denotes  the  volume  of  [„, 

I I5    a  cube,  the  second  power  of  a  number  is     

5  also  called  the  Square,  and  the  third  power  of  a 

number  the  Cube. 

16.  Involution  is  the  process  of  finding  the  power  of  a 

number. 

EXERCISES. 

The  first  power  of  a  number  is  the  number  itself. 

1.  Write  the  first  power  of  the  numbers  represented  by 
the  digits. 

2.  Write  an  equation  to  denote  the  square  of  each  of  the 
following  numbers :  1,  3,  5,  7,  9,  10,  15,  25. 

3.  Write  an  equation  to  denote  the  cube  of  the  following 
numbers :  1,  2,  3,  4,  5,  6,  7,  8,  9,  0. 

4.  Find  the  value  of  x  in  each  of  the  following  equations  : 
x  =  O2,  x  =  I2,  x  =  22,  x  =  32,  x  =  42,  x  =  52,  x  =  62, 
x  =  33,  x  =  43,  x  =  44,  x  =  52,  x  =  53,  x  =  63. 

5.  In  like  manner  show  the  square  of  20,  30,  40,  50,  60, 
70,  80,  90,  100. 

6.  Also,  the  cube  of  10,  20,  30,  40,  50,  60,  70,  80. 

7.  In  this  manner,  (J-)2  =  £,  write  the  square  of  ^-,  J,  £, 


INVOLUTION  BY  ANALYSIS  309 

8.  In  like  manner  write  the  third  power  of  -J-,  -J,  f,  •£. 

9.  Write  the  second  power  of  .1,  .2,  .3,  .4,  .5,  .6,  .7,  .8,  .9, 
in  this  manner:  .1  X  .1  =  .I2  =  .01. 

10.  Find  the  value  of  .I3,  .23,  .33,  .4s,  .5s,  .63,  .73,  .83,  .93. 

11.  Find  the  value  of  (f)2,  (.4)2,  (|)3,  (ff,  ('.!)«,  (.02)3. 

In  the  equation  x  =  42,  the  value  of  x  is  the  square  of  4. 

Process. 

x  =  42  =  4  X  4  —  16. 
In  like  manner  find  the  value  of  x  in  the  following : 

1.  x  =    252.          8.  x  =  103.  15.  x  =  .093. 

2.  x  =    352.          9.  x  =  213.  16.  x  =  .054. 

3.  x  =    882.        10.  x  =  (f)3.          17.  a?  =  .0053. 

4.  a?  =  1012.        11.  x  =  .0013.        18.  x  =  2.052. 

5.  x  =    132.         12.  x  =  .lo3.          19.  a;  =  (25J)2. 

6.  or  =    93.  13.  x  =  .043.          20.  x  =  (4.500f)2. 

7.  x  =    173.        14.  a:  =  .1253.        21.  x  =  (21.65f)2. 

INVOLUTION  BY  ANALYSIS. 

NOTE. — If  this  subject  be  considered  too  difficult,  it  may  be  omitted. 

322  =  (30  +  2)2  =  (tens  +  units)2  =  (t  +  u)2. 
We  will  now  square  32  as  tens  and  units. 

32  =  30  +  2      =  t  +  u 

32  =  30  +  2      =  t  +  u 

22 
30  X  2 


30  X  2  '         '  ""       '  '  '      ' 
302    =    900  — 


1024  =  f  +  2t  x  u  -f  u2. 

Hence  we  have  a  very  important  principle  : 


310  PRACTICAL   ARITHMETIC 

The  square  of  any  number  consisting  of  tens  and  units 
=  the  tens2  f  2  times  the  tens  X  the  units  +  the  units2. 

To  illustrate,  find  the  square  of  35  in  accordance  with  this 
principle. 


300  =  2t  X  u 
352  =  (tens  +  units)2  =  \       25  =  u2 

1225  =  £  +  2t  X  u  +  u2. 

In  like  manner  find  the  square  of  45,  56,  97,  21,  38,  63, 
75,  88,  19,  24. 

We  will  now  proceed  to  find  the  cube  of  32. 

323  =  322  X  321. 

8     =u* 
120 -I 

4^  12ol=3£Xw2 

V    S9-        120' 

=  1800. 

1800  1=  3^  X  u 
1800J 
27000    =$ 


32,768     =  £  +  3*2  X  u  -f  3*  X  u2  +  u*. 
Hence  the  principle : 

The  cube  of  any  number  consisting  of  tens  and  units  = 
the  tens3  +  3  times  the  tens2  X  the  units  -f  3  times  the 
tens  X  the  units2  +  the  units3. 

For  example :  353  =  (30  +  5)3  =  303  +  3  X  302  X  5  •  + 
3  X  30  X  52  +  53  =  27,000  -f  13,500  -f  2250  +  125  == 
42,875. 

In  like  manner  find  the  cube  of  35,  22,  19,  53,  38,  27,  47, 
48,  45,  63,  73,  77. 

Involution,  as  we  have  seen,  develops  the  power  from  the 
root.  The  reverse  process,  that  extracts  the  root  from  the 
power,  is  called  Evolution. 


EVOLUTION  311 

EVOLUTION. 

INDUCTIVE    STEPS. 

1.  The  numbers  represented  by  the  digits  and  their  squares 
are : 

Numbers,  1,     2,     3,     4,     5,     6,     7,     8,     9. 
Squares,      1,     4,     9,    16,  25,  36,  49,  64,  81. 

NOTE. — Pupils  should  memorize  this  table  of  squares  and  roots. 

2.  The  numbers  are  the  square  roots  of  their  squares :  1  is 
the  square  root  of  1  ;  2  is  the  square  root  of  4 ;  3  is  the  square 
root  of  9,  and  so  on. 

3.  A  square  number  is  the  product  of  two  equal  factors, 
either  of  which   is  the  square  root  of  that  square  number. 
64  =  8  X  8 ;    64  is  a  square  number,  and  8  is  its  square 
root. 

4.  Since  27  =  3  X  3  X  3,  27  is  a  cube  number,  and  3  is 
its  cube  root ;  and  since  16  =  2x2x2X2,  16  is  a  fourth 
power,  and  has  2  for  its  fourth  root. 

5.  Evolution  is  the  process  of  finding  the  roots  of  numbers. 
A  number  having  an  exact  root  is  a  perfect  power. 

6.  The  radical  or  root  sign  is  i/        .     The  square  root  of 
64  may  be  thus  expressed  :  1/64,  or  thus  :  64^. 


SQUARE  ROOT. 
By  Factoring-. 

The  prime  factors  of  16  are  2X2  X  2X2  =  4  X  4 ; 
therefore  1/16  =  4.  36  =  2X2  X  3X3  =  6X6;  there- 
fore V/36  =  6. 


312  PRACTICAL   ARITHMETIC 

In  like  manner  find  the  square  root  of: 

1.  144.          5.  576.  9.  1764.  13.  3969. 

2.  196.          6.  676.  10.  1936.  14.  5184. 

3.  256.          7.  1296.          11.  2601.  15.  6400. 

4.  324.          8.  1225.          12.  2916.  16.  8281. 


Periods  and  Roots  Compared. 

Separating  the  following  squares  into  two-figure  periods  as 
far  as  possible,  we  have  : 

One  period.  Two  periods.  Three  periods. 

W  =  1.       l/l '00    =  10.       1/1  WOO    =  100. 


1/81'  =  9.       l/98'Ol  =  99.       1/99'80'01  =  999. 

Obviously,  one  period  in  the  square  gives  but  one  figure  in 
the  root ;  two  periods  in  the  square,  two  figures  in  the  root ; 
three  periods  in  the  square,  three  figures  in  the  root.  Hence 
the  principle : 

The  number  of  figures  in  the  square  root  equals  the 
number  of  two-figure  periods  into  which  the  square  can  be 
pointed  off,  beginning  at  units. 

NOTE. — The  period  on  the  extreme  left  may  contain  but  a  single  figure. 


EXERCISES. 

1.  In  accordance  with  the  foregoing  principle,  state  how 
many  periods  the  squares  of  the  following  roots  contain  :  4, 
9,  32,  99,  317,  999,  3163,  9999,  21,  115,  4156,  19,  316, 
6184,  35,  584,  8196. 

2.  Show  that  the  squares  of  the  numbers  from  4  to  9,  in- 
clusive, give  full  periods. 

Suggestion  :  42  =  16  ;  92  =  81. 


SQUARE  ROOT  313 

3.  Show  that  the  squares  of  the  numbers  from  32  to  99, 
inclusive,  give  two  full  periods. 

4.  Show  that  the  squares  of  the  numbers  from  317  to  999, 
inclusive,  give  three  full  periods. 

5.  Show  that  the  squares  of  the  numbers  from  3163  to  9999, 
inclusive,  give  four  full  periods. 

6.  State  the  exact  number  of  figures  contained  in  the  squares 
of  the  following  numbers :  3,  5,  21,  29,  41,  66,  97,  125,  200. 

7.  Point  off  the  following  numbers  into  periods,  and  tell 
the  number  of  figures  in  the  root  of  each  :  196,  1296,  2809, 
5625,   400,689,   516,961,    182,329,    2:3,804,641,    9,991,921. 

Extraction  of  Square  Root. 

General  Method. 
Find  the  square  root  of  190,969. 

Square.         Root. 

1.  The  number  pointed  off  is 19/09/69(437 

2.  The  greatest  square  in  19  is 16 

3    The  square  root  of  16  is  4  (the  first  figure  of  the 

root) 

4.  The  remainder,  with  the  second  period,  is  ...        309 

5.  Twice  the  root-figure  4  is  8,  the  trial  divisor. 

6.  30  -^  8  gives  3  for  the  second  root-figure. 

7.  3  annexed  to  8  gives  83,  the  complete  divisor. 

8.  83  X  3 249 

9.  The  remainder,  with  the  third  period  annexed,  is  6069 

10.  Twice  43  or  86  is  the  second  trial  divisor. 

11.  606  -=-  86  gives  7  for  the  third  root-figure. 

12.  7  annexed  to  86  gives  867,  the  second  complete 
divisor. 

13.  867  X  7  .  6069 


Therefore  the  exact  root  of  190,969  is  437.  Q 


In  a  treatise  on  arithmetic,  a  scientific  explanation  of  the  square  root  is 
scarcely  admissible,  as  methods  essentially  algebraic  or  geometrical  have  to 
be  adopted. 


314  PRACTICAL   ARITHMETIC 

The  algebraic  discussion,  in  brief,  is  as  follows : 

Let  t  represent  the  tens  of  a  root,  and  u  the  units.  As  we  have  already 
seen  (page  310),  a  square  number  equals  il  -f  2t  X  u  +  u2.  We  will  now 
proceed  to  find  from  this  expression  its  root,  t  -\-  u. 

1.  The  square  is t2  -f  2t  X  u  -f  w2. 

2.  V&  =  t,  the  tens  of  the  root. 

3.  Subtracting ff 

we  have  remaining 2t  X  u  +  w2. 

4.  Dividing  2tf  X  w  by  2t  (twice  the  tens),  we  obtain 
M,  the  units  of  the  root. 

5.  Adding  u  to  the  divisor,  2t,  we  have  2t  -(-  w. 

6.  Multiplying  2t  -f  w  by  w,  we  have 2t  X  M  -f-  M2. 

7.  Subtracting,  we  have  remaining     ....  0 
Therefore  the  root  of  t*  +  2t  X  u  +  «8  is  ^  +  w. 


Brief  directions  are : 

1.  Point  off  the  number  into  two-figure  periods. 

2.  Find  in  the  first  period  the  greatest  square  and  its  root. 

3.  Subtract  and  annex  the  next  period  for  a  remainder. 

4.  Divide  the  remainder  by  twice  this  root  to  find  the 
second  figure  of  the  root. 

5.  Annex  the  quotient  to  both  root  and  divisor. 

6.  Multiply  by  the  units. 

7.  Apply  (3),  (4),  (5),  and  (6)  again,  if  necessary. 

To  apply  the  rule  : 

Find  the  square  root  of  53,361. 

Process. 
Iststep.    .  5'33'61(23i     .    .    .    ?  +  2tXu  +  u>(t  +  u 

'2d  step  .    .  4  .    .    .   1? 

3d  step  .  -v 

4th  step .  |>  43)  133  .    .  2t  +  u)  2t  X  u  +  w2  (+  4) 

5th  step .  J 

6th  step.   .  129  2t  X  u  +  u^ 

7th  step.    .    461  \    461 

The  above  formula  furnishes  two 

figures  of  the  root.     Calling  23  the 
0  tens  of  the  root,  2t  =  46. 


SQUARE  ROOT  315 


EXERCISES. 
Find  the  square  root  of: 


1. 

100. 

11. 

2809. 

21. 

674,041. 

2. 

10,000. 

12. 

3969. 

22. 

784,996. 

3. 

625. 

13. 

4489. 

23. 

944,784. 

4. 

961. 

14. 

7056. 

24. 

998,001. 

5. 

2704. 

15. 

9216. 

25. 

5,875,776. 

6. 

6889. 

16. 

16,129. 

26. 

6,270,016. 

7. 

15,625. 

17. 

70,756. 

27. 

12,574,116. 

8. 

141,376. 

18. 

118,336. 

28. 

30,858,025. 

9. 

160,801. 

19. 

262,144. 

29. 

40,005,625. 

10. 

100,000,000. 

20. 

368,449. 

30. 

29,735,209. 

SQUARE   ROOT   OP   COMMON   AND   DECIMAL 
FRACTIONS. 

INDUCTIVE   STEPS. 
1.  What  is  the  square  root  of  J  ? 


2.  What  is  the  square  root  of  •$•  ? 

=  |;for^X|=4.     Hence.  I/Fraction^ 

V  Denominator 

3.  Find  the  square  root  of  .25. 

1/^25  =  .5;  for  .5  X  .5^.25. 

4.  The  V^i=  what? 

Not  .2,  for  .2  X  .2  =  .04. 
But  V~A  =  V~Ab  =  1/.4000. 


316 


PRACTICAL  ARITHMETIC 


By  rule  :     .40'00  ( .632 

36 

123)400 
369 


Hence 


1262)  3100 
2524 

V~A  =  .632+. 


In  such  cases  annex  ciphers  and  make  periods  from  the 
point  toward  the  right. 


PROBLEMS. 

1. 

Find 

the  square 

root 

of: 

1. 

i 

5. 

TTHT' 

9-  H*i- 

13.  mm- 

2. 

A- 

6. 

TTT- 

10.  HM- 

14.  T*WHHHHb 

3. 

Tthr- 

7. 

AT- 

11.    T^ft 

V 

15.  ifUMf 

4. 

rMHhr- 

8. 

fff 

12.  mn 

*• 

is.  im7^9- 

2. 

Find 

the  square 

root 

of: 

1. 

.09. 

6. 

.12345. 

11. 

.003969. 

2. 

.9. 

7. 

.763876. 

12. 

1.679616. 

3. 

.0144. 

8. 

.30858025. 

13. 

204.7761. 

4. 

.144. 

9. 

.093636. 

14. 

.00009801. 

5. 

.0100. 

10. 

.099225. 

15. 

.00010201. 

3.  Find  the  square  root  of: 

1.  f.  6.  H 

2.  f  7. 

3.  TVW       8-  f 

4-  tfff       9-  I- 
5.  jffy.     10.  f. 


11.  f. 

12.  If. 

13-  I  +  |  +  f 

1/1  a  9 


15. 


16.  f 


18. 
19. 
20. 


Suggestion  :  |  =  .875 ;  i/.ST'SO  =  what? 


SQUARES  317 

SQUARES. 

Since  the  area  of  a  square  lot  whose  side  is  12  rods  equals 
12  X  12  or  144  square  rods,  a  side  of  the  lot  =  T/144. 
Hence  the  formula : 


Side  of  Square  =  v/Area. 

PROBLEMS. 

1.  What  is  the  side  of  a  square  whose  area  is  1225  sq.  It.  ? 

2.  What  is  the  side  of  a  square  whose  area  is  2025  sq.  rd.  ? 

3.  What  is  the  side  of  a  square  farm  containing  40  A.  ? 

4.  A  square  plot  of  ground  contains  320  A.     How  many 
feet  long  is  each  side  ? 

5.  A  circular  pond  has  an  area  of  529  sq.  rd.     What  is 
the  side  of  a  square  of  equal  area  ? 

6.  If  an  acre  of  land  be  laid  out  in  a  square  farm,  what 
will  be  the  length  of  each  side  in  rods  ? 

7.  To  arrange  7225  men  in  the  form  of  a  square,  how 
many  men  must  be  put  in  each  line  ? 

8.  What  would  it  cost  to  fence  a  square  lot  containing 
640  A.  at  $4.00  per  rod  ? 

9.  If  it  cost  $312  to  enclose  a  field  216  rd.  long  and  24 
rd.  wide,  what  will  it  cost  to  enclose  a  square  field  of  equal 
area  with  a  like  kind  of  fence  ? 

10.  The  attempt  to  form  a  square  of  10,200  men  excluded 
200  of  the  men.     How  many  men  stood  in  each  line  of  the 
square  ? 

11.  If  the  faces  of  a  cubical  box  measure  23,064  sq.  in., 
how  many  linear  inches  in  one  of  its  edges  ? 

12.  Which  will  cost  the  more  to  fence,  a  field  measuring 
40  by  80  rd.  or  a  field  of  the  same  area  in  the  form  of  a 
square?     How  much  more  at  $1.33J  per  rod? 


318 


PRACTICAL  ARITHMETIC 


TRIANGLES. 

A  Triangle  is  a  figure  bounded  by  three  straight  lines. 
A  Right  Triangle  has  one  right  angle. 

h  denotes  the  hypotenuse,  the  side  opposite  the 
right  angle  ;  />,  the  perpendicular ;  6,  the  base. 

These  three  lines  are  so  related  that  h2  = 
b2  +  P2- 

Hence  it  follows  that 


l.li  =  i/b2  +  p2 

2.  b  =  i/h2  —  p2 

3.  p  =  l/h'  —  b2 


Formulae. 


Right  Triangle. 

1.  The  base  of  a  right  triangle  is  10  feet,  its  perpendicular 
15  feet.     Find  its  hypotenuse. 

h.  =  1/b2  +  p2  =  1/100  +  225  =  1/326  =  18  very  closely. 

2.  Find  the  sides  indicated 
by  x  in  the  table,  using  formulae 
1,  2,  and  3. 

No.  3.    b  =  Vtf  —  f  = 
~49~=:  1/15  =  3.87  +. 


1 

X 

3 

2 

2 

Q 

X 

5 

3 

8 

i   \    * 

4 

10 

q 

X 

5 

12 

X 

11 

6 

X 

13 

14 

3.  The  perpendicular  of  a 
right  triangle  is  30  ft.  and  the 
hypotenuse  is  50  ft.  What  is 
the  base? 

4.  A  square  floor  contains  400  sq.  ft.     Find  the  length 
of  the  longest  straight  line  that  can  be  drawn  thereon. 

5.  A  tree  150  ft.  high  stood  on  the  bank  of  a  stream.     A 
part  broken  off  125  ft.  from  the  top  exactly  measured  the 
distance  to  the  opposite  banb.     How  wide  was  the  stream  ? 

6.  How  far  from  a  tower  40  ft.  high  must  the  foot  of  a 
ladder  50  ft.  long  be  placed  that  it  may  exactly  reach  the  top 
of  the  tower  ? 


TRIANGLES  319 

7.  The  inner  dimensions  of  a  box  are  36,  24,  and  12. 
Find  the  length  of  the  longest  straight  rod  that  can  be  put 
therein. 

8.  A  ladder  40  ft.  long  is  so  placed  in  a  street  that,  with- 
out being  moved  at  the  foot,  it  will  reach  a  window  on  one 
side  33  ft.  and  on  the  other  side  21   ft.  from  the  ground. 
What  is  the  breadth  of  the  street  ? 

9.  x  =  1/h2  —  b2.     Draw  a  figure  for  this  equation,  and 
write  x  upon  the  line  to  be  found. 

10.  Make  a  ten-foot  pole  the  hypotenuse,  and  find  exact 
lengths  for  the  base  and  perpendicular. 

Three  Sides  Given  to  Find  the  Area. 

1.  If  the  three  sides  of  a  triangle  are  2,  5,  and  6,  what  is 

its  area? 

Process. 

(a.)  *±i±*  =  J£  =  6.5. 

(6.)  6.5  —  2  =  4.5 ;  6.5  —  5  =  1.5 ;  6.5  —  6  =  .5. 

(c.)  Area  =  1/6.5  X  4.5  X  1.5  X  .5  =  1/21.9375  =  4.68. 

Brief  directions  are : 

1.  Find  half  the  sum  of  the  sides. 

2.  From  the  half  sum  subtract  each  side  separately. 

3.  Find  the  square  root  of  the  product  of  the  half  sum 
and  the  three  remainders. 

2.  What  is  the  area  of  a  triangle  whose  sides  are  respect- 
ively 4  in.,  5  in.,  and  6  in.? 

3.  Find  the  area  of  a  triangular  lot  whose  sides  are  respect- 
ively 20,  25,  and  28  rods. 

4.  Find  the  area  of  a  triangular  farm  whose  sides  are  400 
yd.,  500  yd.,  and  600  yd. 

5.  What  is  the  area  of  a  triangle  whose  sides  are  6,  8,  and 
12  ft.? 


320  PRACTICAL  ARITHMETIC 

CUBE  ROOT. 

1.  The  Cube  Root  of  a  number  is  one  of  its  three  equal 
factors.     216  =  6  X  6  X  6;  216  is  therefore  a  cube,  and  6 
is  its  cube  root. 

2.  The  numbers  represented  by  the  digits  and  their  cubes 
are: 

Numbers,  0,  1,  2,     3,     4,       5,       6,       7,       8,       9. 
Cubes,       0,  1,  8,  27,  64,  125,  216,  343,  512,  729. 

NOTE.— This  table  should  be  memorized. 

3.  The  numbers  are  the  cube  roots  of  their  cubes.     1  is 
the  cube  root  of  1 ;  2  is  the  cube  root  of  8 ;  3  is  the  cube 
root  of  27,  and  so  on. 

4.  The  cube  root  of  216  may  be  thus  expressed  :  ^216  or 

Cube  Root  Found  by  Factoring". 


The  prime  factors  of  64  are  2X2X2X2X2X2  = 
4X4X4;  therefore  1^64"=  4. 

216  =  2X2X2  X  3^X~3~X~3  =  6X6X6;  therefore 


In  like  manner  find  the  cube  root  of  27,  125,  343,  512, 
729,  4096,  42,875,  166,375,  185,193. 

Periods  and  Roots  Compared. 

1.  Separating    the    following    numbers    into    three-figure 
periods  as  far  as  possible,  we  have  : 

One  Period.  Two  Periods.  Three  Periods. 

•f]7=  i.         1^1  'ooo  =  10.         fVooo'ooo  ==  100. 


^729"'  =  9.         f980'001  =  99.         1^998'000'001  =  999. 
2.  Obviously  one  period  in  the  cube  gives  but  one  figure 


CUBE  EOOT 


321 


in  the  root ;  two  periods  in  the  cube,  two  figures  in  the  root ; 
three  periods  in  the  cube,  three  figures  in  the  root. 
Hence  the  principle : 

The  number  of  figures  in  the  cube  root  equals  the  num- 
ber of  three-figure  periods  into  which  the  number  can  be 
pointed  off,  beginning  at  units. 

NOTE — The  period  on  the  extreme  left  may  contain  only  one  or  two 
figures. 

Extraction  of  the  Cube  Boot. 

General  Method. 
Find  the  cube  root  of  74,088. 

Cube.    Root. 

1.  The  number,  pointed  off  into  periods,  is '4  Ooo  (  41i 

2.  The  greatest  cube  in  74  is 64 

3.  The  cube  root  of  64  is  4,  the  first  figure  of  the  root. 

4.  Subtracting  and  annexing  the  second  period,  we  have       10088 

5.  300  times  the  root-figure  4*  =  4800,  the  trial  divisor. 

6.  10088  -=-  4800  gives  2  for  the  second  figure  of  the  root. 

7.  The  complete  divisor  consists  of: 

(a. )  The  trial  divisor,     4800 

(b.)  30  times  4  X  2,  or    240 

(c  )  2x2  =  22,  or          _4 

Sum  =  5044 

8.  Multiplying  the  sum,  5044,  by  2,  we  have    ....       10088 

9.  Subtracting,  we  have 0 

Therefore  the  cube  root  of  74,088  is  42. 

The  process,  freed  from  explanation,  stands  thus : 

74'088  ( 42 
64 

4800  10088 
240 
4 

5044  10088 
0 

21 


322  PRACTICAL  ARITHMETIC 

The  algebraic  discussion  is  as  follows  : 

We  have  seen  that  the  cube  of  any  number  consisting  of  tens  and  units 
=  the  tens3  -f  3  times  the  tens2  X  the  units  -f  3  times  the  tens  X  the  units2 
-f  the  units3.  For  example,  353  =  (30  +  5)3  =  303  +  3  X  302  X  5  +  3 
X  30  X  52  +  53.  That  is,  the  cube  of  a  two-digit  number  consists  of  four 
parts,  which  may  be  presented  thus : 


X  u. 

c.  3  X  30  X  52  =  3  X  t  X  u?. 

d.  53  =u*. 


By  regarding  these  four  parts  we  may  readily  see  how  the  cube  root  of 
a  number  may  be  obtained. 

1.  What  is  the  cube  root  of  42,875? 


Process. 


42'875  (  35  Pointing  off,  we  have  :  42'875 

9700      97  Finding  Part  a,  rf3,  we  have  :  27 

'  1^27  =  3,  the  tens  of  the  root. 


A  KA      -i  K 
40U       1  0 

Subtracting  27,  we  have  remaining 

parts6lc,d=  15875 

0  Assume  3  X  V  X  u  =  15875. 


Dividing  by  the  factor  3  X  t2,  we  shall  obtain  the  other  factor, 
the  units. 

3  x  P  =  2700  ;  15,875  -5-  2700  =  5,  the  units  of  the  root. 

Having  thus  found  by  trial  the  units,  we  must  now  form  the 
parts  b,  c,  d,  and  subtract  their  sum. 


X  5  =  3175  X  5  =     15875 
0 

Hence,  the  cube  root  of  42,875  is  3  tens  -f  5  units  =  35. 
The  rule,  briefly  stated,  is  : 

1.  Point  off  the  number  into  three-figure  periods. 

2.  Find  in  the  first  period  the  greatest  cube  and  its  root 


CUBE  BOOT 


323 


3.  Subtract  and  annex  the  second  period. 

4.  To  find  the  second  figure  of  the  root,  divide  the  re- 
mainder by  3OO  times  the  square  of  the  first  root-figure. 

5.  To  this  divisor  add  SO  times  the  product  of  the  two 
root-figures;   also,  the  square  of  the  second  figure. 

6.  Multiply  the  sum  by  the  second  root-figure. 

7.  Then  apply  again  3,  4,  5,  6,  and  7,  if  necessary. 

To  apply  the  rule  : 

2.  Find  the  cube  root  of  79,507. 


Process. 

(1.) 

79'507(43 
(2.)  64  _ 
(3.)  15507 
(6.)  15  507 


4800  (4.) 

360  (5.) 

9  (5.) 

5169(6.) 


0 


3.  Find  the  cube  root  of  2,048,383. 


300  (4.) 
60  (5.) 

4(5.) 

364  (6.) 

proceed  to  find  the  units  by  (4). 

144 

300 

43200  (4.) 

2520  (5.) 

49  (5.) 

45769  (6.) 


Process. 
2'048'383(127,  Ans. 


1048  (3.) 
728  (g.) 
320383  (7.)     Now  call  the  root  12  tens,  and 


320383,  Rem, 
320383 

0(6.) 


324  PKACTICAL  ARITHMETIC 

EXERCISES. 
Find  the  cube  root  of : 

1.  614,125.  8.  2,000,376.  15.  592,704. 

2.  74,088.  9.  153,990,656.  16.  1,860,867. 

3.  15,625.  10.  41,063,625.  17.  34,328,125. 

4.  32,768.  11.  12,167.  18.  145,531,576. 

5.  103,823.          12.  32,768.  19.  264,609,288. 

6.  1,953,125.       13.  79,507.  20.  1,879,080,904. 

7.  5,545,233.       14.  59,319.  21.  12,895,213,625. 

CUBE  ROOT  OP  COMMON  AND  DECIMAL 
FRACTIONS. 

1.  What  is  the  cube  root  of  ^? 

*^  =  i;  foriX  JX  J  =  (J)3  =  3sV. 

FORMULA. 

? Fraction  =     ^Numerator  __ 
v  Denominator 

2.  What  is  the  cube  root  of  .8  ? 

Not  .2,  for  .2  X  .2  X  .2  =  .008. 

&&  =  1^.800000. 
By  rule  :  .SOO'OOO  ( .92  + 

729 

24300  71000 

540  49688 

4  21312,  Rem. 


24844 

Hence  ^8~=  .92  +. 

In  such  cases  annex  ciphers,  and  make  periods  from  the 
point  toward  the  right. 


CUBE  ROOT  OF  COMMON  AND  DECIMAL  FRACTIONS    325 

3.  Find  the  cube  root  of: 

*.  3*.    5!  m-    »!  IPfT 

3-  TJft.       6.  #&.     9.  I- 

4.  Find  the  cube  root  of: 


1.  .008. 

6.  2.197. 

11.  7. 

2.  .08. 

7.  9.261. 

12.  34.965783. 

3.  .8. 

8.  185.193. 

13.  41.063625. 

4.  .125. 

9.  .1. 

14.  .000001. 

5.  .25. 

10.  6. 

15.  .0000001. 

Suggestion  :  1^.08  =  1^.080  =  .4,  etc.     Find  two  more  places. 

5.  Find  the  cube  root  of: 

1.  f.         3.  f       5.  f.       7.  lff|.         9. 

2.  -fy.       4.  f.       6.  f.       8.  ifff.       10. 

Suggestion :  f  =  .75 ;   ^^750  =  what? 

VOLUME. 
Volume  of  a  cube  =  side3.     Therefore  : 


^Volume  =  side  of  cube. 

PROBLEMS. 

1.  A  cubical  cistern  contains  1331  solid  feet.     What  is  the 
length  of  one  side  of  the  cistern  ? 

Volume  =  1331.     1^1331  =  11,  length  of  one  side. 

2.  A  cubical  pedestal  contains  373,248  cu.  in.     What  is 
the  length  of  one  of  its  sides  ? 

3.  A  cubical  box  contains  474,552  cu.  in.     What  is  the 
area  of  one  of  the  surfaces  of  the  box  ? 

4.  How  much  paper  will  cover  the  six  surfaces  of  a  cubical 
box  whose  volume  is   -    cu.  ft.  ? 


326  PRACTICAL   ARITHMETIC 

5.  What  is  the  depth  of  a  cubical   box   that  will  hold 
a  bushel  ? 

6.  A  wagon-box  holds  100  bu.    The  length  is  twice  the  width 
and  the  width  and  depth  are  equal.     Find  the  dimensions. 

7.  Find  the  cost,  at  83  cts.  per  square  yard,  of  lining  the 
inside  of  a  cubical  box  holding  900  gal.  of  water. 

8.  Find  the  height  of  a  cubical  pile  of  wood  containing 
179  cords. 

SIMILAR   FIGURES. 

1.  Similar  figures   have  the   same   shape,  but  differ  in 
size. 

2.  Figures  are  either  surfaces  or  solids. 

3.  A  surface  has  dimensions  and  area. 

4.  A  solid  has  dimensions  and  volume. 

5.  The  relation  of  similar  figures  is  in  accordance  with  the 
following  general  principles : 

Similar  Surfaces. 

1.  The  areas  of  similar  surfaces  are  to  each  other  as  the 
squares  of  their  like  dimensions. 

2.  The  like  dimensions  of  similar  surfaces  are  to  each 
other  as  the  square  roots  of  their  areas. 

PROBLEMS. 

1.  Two  surfaces  having  the  same  shape  are  to  each  other  as 
114  to  36.     What  is  the  ratio  of  their  lengths  ? 

Process. 


L.  :  1.  =  1/144  :  1/36. 
L.  :  1.  =        12:      6. 
L.  :  1.  ==          2  :      1. 

Hence  the  ratio  of  their  lengths  is  2  :  1. 


CUBE  HOOT  OF  COMMON  AND  DECIMAL  FRACTIONS    327 

2.  The  radius  of  a  certain  circle  is  5  ft.    What  is  the  radius 
of  another  circle  containing  twice  the  area  of  the  first? 

Suggestion :  5  :  K.  =  V\  :  1/2. 

3.  The  surfaces  of  two  bodies  having  the  same  shape  are 
as  100  :  25.     What  is  the  ratio  of  their  widths? 

4.  If  the  area  of  a  circle,  whose  diameter  is  2  ft.,  is  6.2832 
sq.  ft.,  what  is  the  diameter  of  a  circle  whose  area  is  25.1328 

sq.  ft? 

Suggestion  :  25.1328  =  4  times  6.2832. 

5.  A  farmer  has  a  field  50  rd.  wide  by  80  rd.  long,  which 
contains  25  A.     Find  the  dimensions  of  a  similar  field  con- 
taining 16.81  A. 

6.  If  a  horse  tied  to  a  stake  by  a  rope  8.79  rd.  long  can 
graze  upon  1J  A.  of  land,  how  long  must  the  rope  be  that  he 
may  graze  upon  6  A .  ? 

7.  If  a  pipe  whose  diameter  is  1.5  in.  fills  a  cistern  in 
5  hours,  in  what  time  will  a  pipe  whose  diameter  is  3  in.  fill 
the  same  cistern  ? 

8.  A  half-inch  pipe  discharges  a  barrel  of  water  in  a  cer- 
tain time.    How  much  will  a  2-in.  pipe  discharge  in  the  same 
time? 

9.  If  a  1-in.  pipe  discharges  1  gal.  in  45  seconds,  how 
much  will  a  2-in.  pipe  discharge  in  60  seconds  ? 

10.  A  rectangular  piece  of  land  has  a  width  of  160  ft.  and 
is  valued  at  $1200.  What  is  the  value  of  a  similar  piece  of 
land  having  twice  the  length  and  breadth  ? 

Similar  Solids. 

1.  The  volumes  of  similar  solids  are  to  each  other  as  the 
cubes  of  their  like  dimensions. 

2.  The  like  dimensions  of  similar  solids  are  to  each  other 
as  the  cube  roots  of  their  volumes. 


328  PRACTICAL  ARITHMETIC 

PROBLEMS. 

1.  Of  two  spheres,  one  is  1000  times  the  size  of  the  other. 
If  the  diameter  of  the  smaller  is  6  inches,  how  many  feet  are 
in  the  diameter  of  the  larger  ? 

Process. 


ri  :  1^1000  =  6  in.  :  x. 

I  :  10  =  6  in.  :  x. 
x  =  10  X  6  =  60  in.  =  5  feet. 

2.  The  diameter  of  a  ball  weighing  32  Ib.  is  6  in.     What 
is  the  diameter  of  a  ball  weighing  4  Ib.  ? 

3.  The  diameters  of  two  spheres  are  respectively  4  and  1 2 
in.      The  larger  sphere  is  how  many  times  the  smaller? 

4.  If  a  2-in.  globe  of  gold  is  worth  $500,  what  is  the 
value  of  a  6-in.  globe  of  gold  ? 

5.  If  the  diameter  of  the  sun  is  112  times  as  long  as  that 
of  the  earth,  how  much  greater  is  the  mass  of  the  sun  than 
that  of  the  earth  ? 

6.  If  the  diameter  of  the  moon  is  2000  mi.  and  that  of 
the  earth  is  8000  mi.,  what  is  the  ratio  of  their  volumes? 

7.  The  weights  of  two  cylinders  of  the  same  shape  are  as 
27  to  64.     What  is  the  ratio  of  their  lengths? 

Process. 

1.  :  L.  =  1^27  :  ^64. 
1.  :  L.  =  3  :  4. 

8.  If  a  log  1J  ft.  in  diameter  contains  35  cu.  ft.,  what  is 
the  diameter  of  a  log  of  the  same  length  that  contains  105 
cubic  feet? 

9.  If  a  pyramid  of  hay  12  ft.  high  contains  8  tons,  how 
high  is  a  similar  pyramid  that  contains  60  tons  ? 


MENSURATION  329 


MENSURATION. 

Mensuration  treats  of  the  measurement  of  lines,  surfaces, 
and  volumes. 

Important  Suggestion. — Experience  has  shown  that  much,  if  not  all,  of 
the  difficulty  in  mensuration  results  from  the  pupil's  failure  fully  to  under- 
stand the  terms  used  in  describing  surfaces  and  solids,  and  from  the  conse- 
quent failure  to  get  a  clear  conception  of  the  objects  themselves.  There- 
fore it  is  suggested  that  pupils  be  required  to  learn  all  definitions.  This 
can  best  be  done  by  a  careful  study  of  the  figures  in  connection  with  the 
definitions.  Concrete  illustration  should  be  used  whenever  possible,  and 
pupils  should  be  permitted  to  handle  objects.  In  the  absence  of  geomet- 
rical forms,  pupils  should  draw  correct  and  neat  figures  to  represent  the 
conditions  of  each  problem.  Time  thus  spent  will  produce  good  results. 


DEFINITIONS. 

1.  A  Line  has  length,  but  no  width. 

2.  A  Straight  Line  is  one  which  has 
the  same  direction  throughout  its  whole 
length.     It  is  the  shortest  distance  be- 
tween two  points. 

3.  A  Curved  Line  is  one  which  changes 
its  direction  at  every  point  in  its  length. 


4.  Parallel     Lines     are     equidistant 
throughout  their  whole  length. 

5.  A  Horizontal  Line  is  a  line  paral- 
lel to  the  horizon.     The  line  A  B  is  hori- 
zontal. 

6.  When  two  straight   lines  meet  or 
intersect  in  such  manner  as  to  form  right 
angles,  they  are  said  to  be  Perpendicular, 
the  one  to  the  other. 


330  PRACTICAL  ARITHMETIC 

7.  A   Vertical   Line    is   one   that   is 
c                   perpendicular  to  the  horizon.      C  0  is  a 

vertical  line. 

8.  An  Angle  is  the  amount  of  diver- 
gence  of  two  lines  which  meet  at  a  point. 

The  point  is  called  the  Vertex.     In  the 
angle  A  0  C,  0  is  the  vertex. 

The  size  of  an  angle  is  not  dependent 
upon  the  length  of  the  lines  which  form 
the  angle. 
9.  There  are  three  kinds  of  angles  : 

1.  Right  Angle. 

2.  Acute  Angle,  less  than  a  right  angle. 

3.  Obtuse  Angle,  greater  than  a  right  angle. 
Draw  an  angle  of  each  kind. 

10.  A  Diagonal  is  a  straight  line  joining  opposite  angles. 

11.  The  Perimeter  measures  the  bounding  line  of  a  sur- 
face. 

12.  An  Inscribed  Figure  is  the  largest  figure  of  a  given 
kind  that  can  be  drawn  within  another.     (See  page  343.) 

13.  A  Circumscribed  Figure  is  the  smallest  figure  of  a 
given  kind  that  can  be  drawn  about  another.      (See  page 
343.) 

14.  Concentric  Circles  are  those  having  the  same  centre. 
The  space  between  two  concentric  circles  is  called  a  Ring. 

Draw  two  concentric  circles. 

SURFACES. 

1.  Surface  is  the  outside  of  anything.     Every  surface  has 
two  dimensions, — length  and  breadth. 

2.  Area  is  the  extent  of  a  surface,  and   is  estimated  in 
square  units ;  as,  square  inches,  square  feet,  square  yards,  etc. 


MENSURATION  331 

3.  A  Plane  Surface  is  flat,  like  the  walls  and  the  floor  of 
the  school-room.     Name  some  plane  surfaces. 

4.  A  Curved  Surface  is  like  that  of  a  ball.     Name  some 
curved  surfaces. 

5.  Surfaces  are  bounded  by  straight  or  curved  lines ;  hence 
the  terms  rectilinear  and  curvilinear  as  applied  to  surfaces. 

TRIANGLES. 

1.  A  Triangle  is  a  plane  surface  having  three  angles  and 
three  sides.  Every  triangle  has  two  dimensions,  altitude  and  base. 


base 

2.  Triangles,  classified  according  to  their  angles,  are  of  three 
kinds : 

1.  Right  Triangle,  having  one  right  angle. 

2.  Obtuse- Angled  Triangle,  having  one  obtuse  angle. 

3.  Acute-Angled  Triangle,  having  three  acute  angles. 
Draw  a  triangle  of  each  kind. 

3.  Triangles  classified  according  to  their  sides  are  of  three 
kinds : 

1 .  Equilateral  Triangle, — all  sides  equal. 

2.  Isosceles  Triangle, — two  sides  equal. 

3.  Scalene  Triangle, — no  two  sides  equal. 
Draw  a  triangle  of  each  kind. 

4.  We  have  learned  (page  164)  that  the  area  of  a  rectangle 
is  the  product  of  the  length  and  breadth  (base  and  altitude). 
Every  triangle  is  regarded  as  one-half  of  a  rectangle  having 
the  same  base  and  altitude ;  hence  the  formula  for  the  area  of 
a  triangle  is : 

Area  of  triangle  =  base  x  altitude 


332 


PRACTICAL  ARITHMETIC 


By  drawing  figures  and  by  cutting  paper  let  pupils  prove 
the  foregoing. 

PROBLEMS. 

1.  The  base  of  a  triangle  is  150  yd.  and  its  altitude  is  75 
yd.     What  is  its  area? 

2.  Required  the  area  of  a  triangle  whose  base  is  40  rd.  and 
altitude  30  rd. 

3.  What  is  the  area  of  an  equilateral  triangle  whose  sides 
are  each  10  chains? 

4.  A  board  5  ft.  long  has  the  shape  of  an  isosceles  triangle 
and  measures  at  its  base  15   inches.     Find  the  number  of 
square  feet  it  contains. 

5.  Find  the  area  of  a  right  triangle,  base  23.1  ft.,  altitude 
32.1  ft. 

PARALLELOGRAM. 


alt. 


Rhomboid. 


1.  A  Parallelogram  is  a 
plane  surface  whose  opposite 
sides  are  parallel. 

2.  There  are  four  paral- 
lelograms : 

1.  Square — Sides   parallel    and    equal;    four    right 

angles. 

2.  Oblong — Sides  parallel ;  opposite  sides  equal,  ad- 

jacent sides  unequal ;  four  right  angles. 

3.  Rhombus — Sides  parallel  and  equal ;  two  angles 

obtuse  and  two  acute. 

4.  Rhomboid — Sides  parallel ;  opposite  sides  equal ; 

two  angles  obtuse  and  two  acute. 

3.  The  altitude  of  the  Rhombus  and  the  Rhomboid  is  the 
perpendicular  distance  between  the  parallel  sides. 

4.  Make  correct  forms  of  the  parallelograms.     Draw  the 
diagonal  and  mark  the  altitude. 


MENSURATION  333 

5.  The  formula  for  the  area  of  a  parallelogram  is  : 
Area  =  base  X  altitude. 

PROBLEMS. 

1.  A  field  in  the  form  of  a  square  is  64  rd.  long.     Find  its 
area  in  acres. 

2.  How  many  square  feet  in  an  oblong  board  90  in.  long 
and  14  in.  wide? 

3.  A  pane  has  the  form  of  a  rhombus,  measures  16  in.  on 
each  side,  and  the  perpendicular  distance  between  its  sides  is 
one-half  the  length  of  a  side.     Find  its  area. 

4.  Find  the  area  in  acres  of  a  rhomboidal  field  which  meas- 
ures 10  ch.  in  length  and  8  ch.  in  breadth. 

TRAPEZOID. 


1.  A  Trapezoid  is  a  four- 


sided  plane  figure  having  two         /  ,. 

sides  parallel. 

2.  The  altitude  of  a  trape- 


zoid  is  the  perpendicular  distance  between  the  parallel  sides. 
3.  The  formula  for  the  area  of  a  trapezoid  is  : 

Area  =  gum^Qf_parallel  sides  x  altitude. 

What  do  you  get  when  you  divide  the  sum  of  the  parallel 
sides  by  2  ? 

PROBLEMS. 

1 .  Find  the  area  of  a  trapezoid  with  parallel  sides  of  50  rd. 
and  78  rd.,  and  with  a  distance  between  them  of  39J  rd. 

2.  A  trapezoidal  field  contains  12J  A.     Its  parallel  sides  are 
220  rd.  and  180  rd.     How  far  apart  are  the  parallel  sides? 


334  PRACTICAL   ARITHMETIC 

THE    TRAPEZIUM. 

1.  A  Trapezium  is  a  four- 
sided  plane  surface  having  no 
two  of  its  sides  parallel. 

2 .  The  diagonal  of  a  trape- 
zium is  a  straight  line  con- 
necting opposite  angles.     The 
diagonal    divides    the   trape- 
zium into  two  triangles. 

3.  The  altitude  of  each  triangle  is  the  perpendicular  dis- 
tance between  the  diagonal  and  the  opposite  angle. 

4.  To  find  the  area  of  a  trapezium,  the  diagonal  and  the 
altitude  of  each  triangle  being  given,  first  find  the  area  of  each 
triangle,  then  add  the  areas. 

5.  The  following  is  the  formula  : 

Area  =  diagonal  x  *um  of  Attitudes. 

2 

PROBLEMS. 

1.  A  field  has  the  form  of  a  trapezium  with  a  diagonal 
length  of  1000  ft.,  and  with  perpendicular  distances  of  450 
and  350  ft.     Find  the  area. 

2.  Require  the  area  of  a  trapezium  whose  diagonal  meas- 
ures 145  ft.  and  the  altitudes  of  the  two  triangles  are  34  and 
44  ft.  respectively. 

THE    REGULAR   POLYGON. 

1.  Every  plane  surface  bounded  by  straight  lines  has  as 
many  angles  as  it  has  sides. 

Naming  plane  figures  according  to  the  number  of  angles 
each  contains,  we  have  the  following : 


MENSURATION  335 

Triangle,  three  angles;  Quadrangle,  four  angles;  Pen- 
tangle,  or  Pentagon,  five  angles;  Hexagon,  six  angles; 
Heptagon,  seven  angles;  Octagon,  eight  angles;  Nonagon, 
nine  angles;  Decagon,  ten  angles,  etc. 

2.  Polygon  is  a  general  term,  and   is 
applicable  to  any   figure  having   three  or 
more  angles. 

3.  A  Regular  Polygon  is  one  having 
all  its  angles  and  sides  equal. 

4.  Any  regular  polygon  may  be  divided  Hexagon, 
into  as  many  equal  triangles  as  the  polygon  has  sides.     If  the 
base  and  the  altitude  of  the  triangles  be  known,  the  area  of 
the  polygon  may  be  found  by  multiplying  the  area  of  one 
triangle  by  the  number  of  triangles. 

5.  The  formula  for  the  area  of  a  regular  polygon  is : 

Area  =  perimeter  X  Perpendicular 

2 

NOTE. — The  word  "perpendicular''  is  here  used  to  denote  the  altitude 
of  one  triangle. 

PROBLEMS. 

1.  Find  the  art-a  of  a  hexagon  whose  sides  are  each  12  in. 
and  the  perpendicular  distance  from  the  centre  to  a  side  is  8  in. 

2.  What  is  the  area  of  a  regular  pentagon  whose  side  is  15 
ft.  and  the  altitude  of  the  triangles  into  which  it  may  be 
divided  is  8.602  ft.  ? 

THE    CIRCLE. 

1.  A  Circle  is  a  plane  surface 
bounded  by  a  curved  line  every  point 
of  which  is  equally  distant  from  a 
point  within  the  circle  called  the  centre. 
The  point  where  the  straight  lines  in 
the  figure  meet  is  the  Centre  of  the 
circle. 


336  PRACTICAL  ARITHMETIC 

2.  The  Circumference  of  a  circle  is  the  bounding  line. 

3.  The  Diameter  is  the  distance  across  the  circle  measured 
through  the  centre. 

4.  The  Radius  is  one-half  of  the  diameter. 

By  a  geometrical  process,  it  has  been  found  that  if  the  di- 
ameter of  a  circle  is  1,  the  circumference  is  3.1416.  Hence, 
if  we  know  the  diameter  of  a  circle,  we  may  find  the  circum- 
ference by  multiplying  the  diameter  by  3.1416  ;  and  knowing 
the  circumference,  we  may  find  the  diameter  by  dividing  the 
circumference  by  3.1416. 

The  number  3.1416  is  called  the  ratio  of  a  circumference 
to  its  diameter.  Pupils  should  remember  this  number,  as  it 
is  of  much  use  in  measuring  circular  surfaces,  etc. 

5.  The  following  formulae  apply  to  the  circle  : 

1.  Circumference  =  diameter  X  3.1416. 

2.  Diameter  =  gkg|niference 

3.  Area  =  circumference  X  I^l~f*~. 

4.  Area  =  Radius2  X  3.1416. 

5.  Area  =  diameter2  X  .7854. 

.—  Observe  that  .7854  is  one-fourth  of  3.1416. 


MISCELLANEOUS  PROBLEMS. 

1.  What  is  the  circumference  of  a  circle  having  a  diameter 
of  21  ft.  ? 

2.  What  is   the  diameter  of  a  circle  33  yd.  in  circum- 
ference ? 

3.  What  is  the  circumference  of  a  circle  whose  radius  is 
16yd.? 

4.  What  is  the  area  of  a  circle  whose   circumference   is 
18  in.? 

5.  Find  the  perimeter  of  a  triangle  whose  sides  are  re- 
spectively 3J  ft.,  4f  ft.,  and  5{  ft. 


MENSURATION  337 

6.  A  horse  is  tied  by  a  rope  7  rd.  long,  and  can  reach  2  ft. 
beyond  the  end  of  the  rope.     How  much  surface  can  he  graze 
over? 

7.  Find  the  circumference  of  a  circle  whose  diameter  is 
14  ft. 

8.  Find  the  diameter  of  a  circle  whose  circumference  is 
1  ft, 

9.  Find  the  area  of  a  circle  whose  radius  is  7  yd. 

10.  The  radius  of  a  grass  plot  is  42  ft.     Find  the  area  of 
a  walk  4  ft.  wide  running  around  the  grass  plot. 

11.  Find  the  area  of  a  triangle  whose  base  is  10  ft.  and 
altitude  2|  ft. 

12.  What  is  the  area  of  a  trapezium  the  diagonal  of  which 
is  1 1 0  ft.,  and  the  perpendiculars  to  the  diagonal  are  40  ft. 
and  60  ft.  respectively  ? 

13.  If  a  horse  is  tethered  by  a  rope  20  rd.  long,  over  how 
much  surface  can  he  graze  ? 

14.  The  base  of  a  triangle  is  300  yd.  and  its  altitude  is  150 
yd.     Find  the  area. 

15.  Two  opposite  sides  of  a  quadrangular  field  are  parallel, 
and  are  140  yd.  and   170  yd.  long.     The  shortest  measure 
across  the  field  is  90  yd.     What  is  the  area? 

16.  A  rectangular  tank  is  12  ft.  long,  4  ft.  wide,  and  3  ft. 
high.     How  many  square  feet  of  sheet  lead  will  be  required 
to  line  it? 

17.  A  diagonal  of  a  field  in  the  form  of  a  trapezium  is  17 
chains  56  links ;  the  perpendiculars  to  that  diagonal  from  the 
opposite  angles  are  8  chains  82  links,  and  7  chains  73  links. 
What  is  the  area? 

18.  Find  the  diameter  of  a  circle  whose  circumference  is 
316  ft. 

19.  What  is  the  circumference  of  a  circular  pond  whose 
diameter  is  45  rods  ? 

22 


338  PRACTICAL  ARITHMETIC 

20.  What  is  the  area  in  acres  of  a  circular  island  whose 
circumference  is  2  miles  ? 

21.  A  farm  in  the  form  of  a  trapezoid  has  its  parallel  sides 
72  ch.  and  84  ch.  in  length,  and  the  perpendicular  distance 
between  them  is  40  ch.     How  large  is  the  farm  ? 

22.  How  many  rods  of  fence  will  be  needed  to  go  round  a 
circular  park  containing  120  A.? 

Suggestion :  Draw  figures  to  illustrate  the  following  problems. 

23.  A  circular  yard  200  feet  in  diameter  has  a  walk  6  feet 
wide  bordering  on  the  circumference  and  extending  entirely 
around  the  yard.     What  is  the  area  of  the  walk? 

24.  If  within  a  circle  10  feet  in  diameter  a  circle  6  feet  in 
diameter  be  drawn  so  that  the  two  circles  shall  meet  at  one 
point,  what  will  be  the  area  of  the  crescent  thus  formed  ? 

25.  The  side  of  the  largest  regular  hexagon  that  can  be  in- 
scribed within  a  circle  6  ft.  in  diameter  is  equal  to  the  radius 
of  the  circle.     How  much  waste  will  there  be  in  cutting  such 
hexagon  from  the  circle? 

26.  After  making  the  hexagon  in  problem  25,  suppose  you 
should  decide  to  make  from  the  hexagon  as  large  a  circle  as 
possible,  what  would  be  the  diameter  of  the  circle  ? 

VOLUMES. 

1.  A  Solid  has  three  dimensions,  length,  breadth,  and  thick- 
ness. 

2.  The  Volume  of  a  solid  is  the  number  of  cubic  units 
which  it  contains ;  it  may  be  cubic  inches,  cubic  feet,  etc. 

3.  The  Lateral  Surface  of  a  solid  is  the  area  of  its  sides 
or  faces.     This  is  also  called  Convex  Surface. 

4.  To  find  the  volume  of  a  solid  three  dimensions  or  their 
equivalent  must  be  given ;  and  to  find  any  one  of  the  dimen- 
sions of  a  solid  the  volume  and  two  dimensions  or  their 
equivalent  must  be  given. 


MEJXSUKATION 


339 


THE    PRISM    AND    CYLINDER. 


1.  Prism    is   a  solid    whose   ends   are   equal 
parallel  polygons,  and  whose  sides  are  rectangles. 
The  ends  are  called  bases  and  the  sides  are  called 
lateral  faces. 

2.  The  form  of  the  base  gives  a  prism   its 
distinguishing  name.     If  the  base  be  a  triangle, 
the  prism   is  called  a  triangular  prism;  if  the 
base  be  a  square,  the  prism  is  called  a  square 

prism  ;  if  the  base  be  a  pentagon,  the  prism  is  called  a  pent- 
angular prism,  etc. 

3.  A  Cylinder  is  a  solid  with  circular  ends 
and  uniform  diameter.     The  ends  are  called  the 
bases,  and  the  curved  surface  is  called  the  lateral 
surface,  or  convex  surface. 

4.  The  following  formula  apply  to  prisms  and 
cylinders : 

1.  L.  S.  =  Perimeter  of  Base  X  Altitude. 
2.  Vol.  =  Area  of  Base  X  Altitude.  Base  a  circle 


PROBLEMS. 

1.  Find  the  lateral  surface  of  a  pentangular  prism,  the 
side  of  the  base  being  8  in.  and  the  height  35  in. 

2.  Find  the  lateral  surface  of  a  cylinder  whose  height  is 
25  in.  and  diameter  of  the  base  15  in. 

3.  Find  the  lateral  surface  of  a  triangular  prism    24    ft. 
high,  the  sides  of  the  base  being  3  ft.,  4  ft.,  and  5  ft. 

4.  Find  the  entire  surface  of  a  cylinder  9   ft.  high  and  3 
ft.  in  diameter. 

5.  Find  the  entire  surface  of  a  prism  18  in.  square  and  7 
ft.  high. 


340 


PRACTICAL  ARITHMETIC 


6.  Find  the  entire  surface  of  a  prism  18  in.  high,  the  base 
being  a  triangle  whose  sides  are  3  in.,  4  in.,  and  4J  in. 

7.  Estimate  the  volumes  of  the  solids  described  in  prob- 
lems 2,  3,  4,  5,  and  6. 

8.  What  must  be  the  diameter  of  a  cylindrical  tank  10 
ft.  deep  to  contain  8460.288  gal.  ? 

9.  A  rectangular  bin  5  ft  4  in.  long  and  3  ft.  2  in.  wide 
contains  64  bu.     What  is  the  depth  ? 

10.  If  you  cut  a  cylinder  as  large  as  can  be  made  from  a 
prism  6  in.  square  and  18  in.  long,  how  much  of  the  prism 
will  be  wasted  ? 

THE    PYRAMID    AND    CONE. 

1.  A  Pyramid  is  a  solid  having  a  regular  polygon  for  a 
base  and  ending  in  a  point  at  the  top. 

Draw   a   triangular   pyramid.      A    square 
pyramid. 

2.  A   Cone   is  a  solid  having  a  circular 
base  and  tapering  to  a  point. 

3.  The  point  of  a  pyramid  and  of  a  cone  is 
called  the  Vertex. 

4.  The  Altitude  of  a  cone  and  of  a  pyra- 
mid is  a  straight  line  drawn  from  the  vertex 
perpendicular  to  the  base, 

5.  The  Slant  Height  of  a  pyramid  is  a 
straight  line  drawn  from  the  vertex  perpen- 
dicular to  one  side  of  the  base,  as  A  B. 

6.  The  Slant  Height  of  a  cone  is  a  straight 
line  drawn  from  the  vertex  to  any  point  on 
the  circumference  of  the  base. 

Slant  Height 


Base  a  circle 


1.  L.  S.  =  Perimeter  of  Base  X 

2.  Vol.  =  Area  of  Base  X  Altitude 

3 


MENSURATION 


341 


PROBLEMS. 

1 .  Find  the  lateral  surface  of  a  hexagonal  pyramid  whose 
slant  height  is  20  ft.  and  each  side  of  the  base  5  ft. 

2.  What  is  the  extent  of  the  lateral  surface  of  a  cone  the 
base  of  which  is  27  in.  in  diameter  and  the  slant  height  5  ft.  ? 

3.  If  wheat  be  piled  in  a  corner  of  a  rectangular  room  in 
such  manner  as  to  form   a  portion  of  a  cone,   how  many 
bushels  are  in  the  pile  if  the  top  of  the  pile  is  8  ft.  from  the 
floor  and  the  outer  edge  5  ft.  from  the  angle  formed  by  the 
walls  ? 

4.  A  conical  glass  is  7  in.  deep  and  5J  in.  in  diameter. 
What  part  of  a  gallon  will  it  hold? 

5.  A  quadrangular  pyramid  is  16  in.  square  at  the  base 
and  3  ft.  high.     In  making  from  this  pyramid  the  largest 
possible  cone,  how  much  must  be  cut  off? 


FRUSTUM  OF  PYRAMID  AND  OF  CONE. 

1.  A  Frustum  of  a  Pyramid  is  that 
part  of  a  pyramid  which  remains  when  the 
top  is  cut  off  by  a  plane  parallel  to  the  base. 

2.  A  Frustum  of  a  Cone  is  that  part  of 
a  cone  which  remains  when  the  top  is  cut 
off  by  a  plane  parallel  to  the  base. 

3.  The  Altitude  and  the  Slant  Height 
of  frustums  are  found  in  the  same  manner 
as  in  the  case  of  the  pyramid  and  the  cone. 

4.  The  following  formulae  are  applicable 
to  pyramids  and  cones : 


1.  L.  S  = 


Sum  of  Perimeters  of  the  2  Bases 
2 


2.  Vol.  =  [Sum  of  Bases  +  V Product  of  Bases]  X 


X  Slant  Height. 
Altitude 


342  PEACTICAL  ARITHMETIC 

PROBLEMS. 

1.  Find  the  lateral  surface  of  a  frustum  of  a  pentangular 
pyramid  if  the  side  of  the  lower  and  upper  bases  be  3  ft.  and 
2  ft.,  respectively,  and  the  slant  height  9  ft. 

2.  What  is  the  entire  surface  of  a  frustum  of  a  cone,  the  bases 
being  16  in.  and  10  in.  in  diameter  and  the  altitude  -30  in. 

Suggestion  :  First  find  the  slant  height. 

3.  What  is  the  volume  of  the  frustum  described  in  the 
second  problem? 

4.  Find  the  volume  of  a  frustum  of  a  pyramid  4  J  ft.  square 
at  the  lower  base,  2J  ft  square  at  upper  base,  and  6J  ft.  high. 

5.  At  $1.25  a  square  foot  what  will  be  the  cost  of  lining 
with  copper  a  vat  in  the  shape  of  an  inverted  frustum  of  a 
cone  if  the  upper  diameter  is  7  ft.,  the  lower  diameter  5  ft., 
and  the  depth  6  ft.  ? 

THE    SPHERE. 

1.  A  Sphere  is  a  solid  bounded  by  a  curved  surface  of 
which  every  point  is  equally  distant  from  a  point  called  the 
centre. 

2.  The   following   formulae  are   for  the 
surface  and  volume  of  a  sphere : 

1.  Sur.  =  Diameter  X  Circumference. 

2.  Sur.  =  Diameter2  X  3.1416. 

3.  Vol.  =  Sur.   X   |p 

4.  Vol.  =  Diameter3  X  .5236. 

NOTE.— .5236  is  one-sixth  of  3.1416. 

PROBLEMS. 

1.  What  is  the  surface  of  a  sphere  18   in.  in  diameter? 
Its  volume? 

2.  The  diameter  of  a  sphere  is  12  in.,  the  circumference  is 
37.6992  in.     What  is  the  surface? 


MENSURATION  343 

3.  What  is  the  volume  of  a  sphere  the  surface  of  which  is 
78.54  sq.  in.  and  the  radius  is  2.5  in.  ? 

4.  If  the  diameter  of  a  cannon-ball  is  15  in.,  what  is  the 
volume  ?     What  is  the  surface  ? 

5.  A  hemispherical  bowl  12  in.  in  diameter  is  filled  with 
water.     An  iron  ball  put  into  the  water  is  just  large  enough 
to  extend  from  the  bottom  of  the  bowl  to  the  surface  of  the 
water.     Find  the  amount  of  water  that  remains  in  the  bowl 
after  the  sinking  of  the  ball. 

CIRCLE   AND    LARGEST    SQUARE. 
h  is  obviously  both  diameter  of  the  circle  and  hypotenuse  of 
a  right  triangle  ;  b  and  p  are  base  and 
perpendicular,    and    also    sides    of    the 
square.     Since  6  =  -  p,  h2  =  262.      Let 
h  =  10  ;  then  262  =  100,  and  62  =±  1M-. 
Taking  the  square  root,  we  have  b  = 
Hence  the  formula  : 


Side  of  square  = 


When  the  diameter  =  1,  the  side  of  the  square  =  v\  or  .5 
=  .7071  4~?  and  the  formula  becomes  : 

Side  of  square  =  diameter  X  .7O71. 

PROBLEMS. 

1.  If  6  and  p  each  equal  1  (see  figure),  what  is  the  length 
of  A? 

2.  If  p  or  b  equal  1,  what  is  the  length  of  the  circum- 
ference ? 

3.  When  the  diameter  of  a  circle  equals  5,  what  is  the  side 
of  the  inscribed  square? 

4.  Find  the  area  of  the  inscribed  square  and  of  the  circum- 
scribed circle,  when  the  diameter  equals  5. 


344  PRACTICAL  ARITHMETIC 

SPHERE    AND    LARGEST    CUBE. 

hf  is  obviously  both  diameter  of  the  sphere  and  hypotenuse 
of  the  erect  right  triangle,  ///,  7i,  p-}  h 
is  the  hypotenuse  of  the  horizontal  tri- 
angle, h,  b,  b.  h2  =  2b2.  (hj  =  h2 
-f  p2.  Hence  (hj  ==  2b2  +  p2.  But 
b  =  p;  therefore  (h')2  =  362.  Let  h' 
=  10 ;  then  362  =  100,  and  b2  =  ^. 
Taking  the  square  root,  we  have  6  = 
Hence  the  formula : 


Side  of  Cube  =     /diameter' 


When  the  diameter  =  1,  the  side  of  the  cube  =  V \  or  .3333, 
etc.  —  .57735  -f-,  and  the  formula  becomes : 

Side  of  cube  =  diameter  x  .57735. 

PROBLEMS. 

1.  What  is  the  volume  of  a  pyramid  whose  base  is  a  rec- 
tangle 13  by  14  feet,  and  whose  height  is  18  feet? 

2.  What  is  the  volume  of  a  cylinder  108  in.  in  diameter 
and  10  ft.  long? 

3.  What  is  the  lateral  surface  of  a  cone  whose  base  is  10 
ft.  in  diameter  and  slant  height  20  ft.  ?     Find  also  the  entire 
surface. 

4.  Find  the  surface  of  a  sphere  whose  radius  is  12  inches. 

5.  How  many  gallons  will  a  hollow  globe  contain  whose 
inside  diameter  is  20  inches  ? 

6.  What  is  the  lateral  surface  of  a  triangular  prism  whose 
sides  are  each  6  feet  and  whose  altitude  is  8  feet  ? 

7.  What  is  the  lateral  surface  of  a  quadrangular  pyramid 
whose  base  is  15  feet  square  and  the  slant  height  18  feet? 


MENSURATION  345 

8.  What  is  the  lateral  surface  of  a  cone  whose  base  is  10 
ft.  in  diameter  and  whose  slant  height  is  10  ft.  ? 

9.  Find  the  volumes  in  problems  No.  6,  7,  and  8. 

10.  Required  the  surface  of  the  frustum  of  a  cone  whose 
slant  height  is  12  feet,  diameter  of  lower  base  10  ft.  and  upper 
base  6  feet.     What  is  the  volume? 

11.  Find  the  entire  surface  of  the  frustum  of  a  triangular 
pyramid  whose  slant  height  is  40  in.,  and  the  sides  of  the 
upper  base  4  in.  and  the  lower  base  10  in. 

12.  Required  the  contents  of  a  cannon  ball  whose  diameter 
is  9  inches.     What  is  the  surface  ? 

13.  At  45  cents  a  square  foot,  how  much  will  it  cost  to  gild 
a  ball  25  inches  in  diameter? 

14.  Find  how  many  cubic  inches  of  iron  there  are  in  a 
hollow  sphere,  the  diameter  being  15  inches  long  and  the  shell 
3  inches  thick  ? 

15.  A  cylindrical  can  is  6   inches  deep  and  4  inches  in 
diameter.     If  a  cone  of  the  same  height  and  diameter  be 
placed  in  the  can,  how  much  water  will  be  required  to  fill  the 
remaining  space? 

16.  In  the  above  problem,  what  is  the  ratio  of  the  volume 
of  the  cone  and  cylinder?     Does  this  show  why  3  is  used  in 
the  formula  for  the  volume  of  a  cone? 

17.  Find  the  side  of  the  greatest  square  that  can  be  in- 
scribed in  a  circle  whose  diameter  is  10  feet? 

18.  Find  the  edge  of  the  greatest  cube  that  can  be  cut-from 
a  wooden  ball  whose  diameter  is  5.5  inches. 

19.  I  have  a  cubical  box  whose  faces  each  contain  64  square 
inches.     Find  the  diameter  of  the  sphere  that  will  exactly 
contain  the  box. 

20.  I  have  a  circular  garden  whose  circumference  is  31.416 
rods.     I  wish  to  reduce  it  within  the  circumference  to  the 
largest  possible  square  form.     Find  the  area  of  the  square. 


346  PEACTICAL  AKITHMETIC 

GENERAL   REVIEW. 

The  following  problems  have  been  selected  from  the  ex- 
amination papers  of  the  University  of  the  State  of  New 
York. 

They  are  introduced  here  for  the  purpose  of  affording  a 
complete  review  of  the  principles  and  methods  set  forth  in 
the  previous  pages  of  the  book. 

It  is  suggested  that  the  best  efforts  of  both  teacher  and 
pupil  be  applied  to  these  problems,  and  that  the  science  and 
art  of  arithmetic,  as  already  illustrated,  be  faithfully  recalled, 
studied  afresh,  and  securely  fixed  in  mind. 

Let  every  solution,  therefore,  proceed  systematically,  and 
every  principle  involved  be  distinctly  stated. 

1.  Define  sum,  and  illustrate  your  definition  by  a  practical 
example. 

2.  A  man  deposits  in  bank  $986.46.     At  different  times 
he  has  drawn  the  following  amounts  :  $314.18,  $49.25,  $57.62, 
$39.84,  $25.13.     Find  the  amount  remaining  in  the  bank. 

3.  Find  the  least  number  of  bushels  of  grain  that  can  be 
exactly  measured  either  by  a  3-quart,  a  peck,  a  20-quart,  or  a 
bushel  measure. 

4.  Reduce  M-JI4  to  its  lowest  terms. 

&  £t  y  o  / 

5.  Simplify     *  _  .         and  express  the  result  both  as  a 

common  and  as  a  decimal  fraction. 

6.  Define  composite  number  and  give  an  example. 

7.  Make  a  receipted  bill  for  the  following :  Harold  Kirby 
bought  of  Pliny  Hall,  10  Ib.  sugar  at  5  cts.,  J  Ib.  tea  at  60 
cts.,  3  Ib.  coffee  at  40  cts.,  1  sack  flour  at  $1.50. 

8.  If  the  shadow  of  a  post  6  ft.  high   is   4  ft.  6  in.  long, 
what  is  the  height  of  a  tree  whose  shadow  at  the  same  time  is 
125  ft.  long?     (Solve  by  analysis.) 


GENERAL    KEVIEW  347 

9.   What  would  it  cost  to  dig  a  cellar  80  ft.  X  35  ft.  X  8 
ft.  at  $.84  per  cubic  yard  ? 

10.  A  railway  train  runs  f  of  a  mile  in  f  of  a  minute. 
Find  its  velocity  per  hour  ?     (Solve  by  analysis.) 


11.  Define  quotient,  and  give  an  illustration. 

12.  Find  the  prime  factors  of  1001  and  1309,  and  from 
these  factors  form  the  G.  C.  D.,  and  the  L.  C.  Dd.  (least 
common  multiple)  of  the  two  numbers. 

13.  A  field  10  chains  50  links  long  and  8  chains  40  links 
wide  produces  40  bushels  of  oats  per  acre ;  what  is  the  value 
of  the  crop  at  35  cents  a  bushel  ? 

14.  Find  the  sum  of  9f,  8J,  5f,  and  ^-.     Express  the  re- 
sult both  as  a  fraction  in  lowest  terms  and  as  a  decimal. 

15.  What  part   of  an   ounce  (apothecaries'  weight)  is   5 
drachms  and  2  scruples  ? 

16.  Find  the  cost  of  a  stick  of  timber  40  ft.  long,  12  in. 
wide,  9  in.  thick,  at  $12.50  per  M.,  board  measure. 

17.  A  roll  of  wall  paper  8  yd.  long  and  18  in.  wide  costs 
25  cts.     What  will  be  the  cost  of  paper  for  the  four  walls  of 
a  room  30  ft.  X  27  ft.  X  9  ft.,  no  allowance  being  made  for 
openings  ? 

18.  I  bought  240  barrels  of  apples  at  $1.75  a  barrel ;  lost 
40  barrels  through  frost ;  at  what  price  a  barrel  must  I  sell 
the  remainder  to  gain  25%  on  the  money  invested? 

19.  If  2  men  plough  15  acres  in  5  days,  working  10  hours 
a  day,  how  many  acres  will  3  men  plough  in  4  days,  working 
8  hours  a  day  ? 

20.  Define  greatest  common  divisor  and  least  common  divi- 
dend (multiple).     Illustrate. 


348  PRACTICAL  ARITHMETIC 

21.  What  is  meant  by  cancellation  f 

22.  Simplify  J  of  -^-  of  2J  X  14. 

23.  What  part  of  a  bushel  is  contained  in  a  rectangular 
box  3  in.  deep  and  4  in.  square?      [A  bushel  —  2150.42 
cu.  in.] 

24.  From  sixty  subtract  forty-seven  and  sixteen  ten-mil- 
lionths  and  express  the  decimal  as  a  common  fraction. 

25.  Find  the  cost  of  carpeting  a  room  18  ft.  long,  15  ft. 
wide,  with  carpet  27  in.  wide,  at  75  cts.  a  yard. 

26.  Define  divisor,  root,  proportion,  fraction. 

27.  I  retail  oranges  at  3  cts.  each,  gaining  150%  on  the 
purchase  price.     What  did  the  oranges  cost  a  dozen? 

28.  I  sell  an  article  at  an  advance  of  25%  on  the  cost  and 
then  discount  the  bill  5%  for  cash  payment.     My  net  gain  is 
$63.75.     Find  the  cost, 

29.  A  cubic  foot  of  water  weighs  62J  Ib.     Find  the  weight 
of  a  barrel  of  water. 

JJ0T  On  a  bill  of  goods  amounting  to  $485.50  I  receive 
commercial  discounts  of  15%,  10%,  and  5%.  Find  the  net 
cost  of  the  goods. 

31.  What  principal  loaned  for  1  yr.  and  3  mo.  at  6%  simple 
interest  will  amount  to  $1000? 

32.  A  30-day  note  discounted  at  a  New  York  bank  yields 
$358.02.     What  was  the  face  of  the  note? 

33.  A  note  for  $500  at  90  days,  with  interest  at  6%,  is 
discounted  at  a  bank   30  days  after  it  is  dated.     Find  the 
proceeds. 

34.  A  certain  stock   pays  annual   dividends  of  4%.     At 
what  rate  must  it  be  bought  to  pay  5%  on  the  investment? 

35.  Find  the  square  root  of  4,004,231  to  two  places  of 
decimals. 


GENERAL    REVIEW  349 

36.  If  I  buy  10  shares  of  railway  stock  at  80  and   sell 
them  at  90,  how  many  dollars  do  I  gain  and  what  is  the  rate 
per  cent,  of  profit  ? 

37.  Find  the  smallest  number  that  will  exactly  contain  15, 
18,  21,  24,  and  30. 

38.  Two  men  hire  a  pasture  for  $30.     A.  puts  in  8  horses 
for  10  weeks  and  B.  6  horses  for  12  weeks.     How  much 
should  each  pay? 

39.  A  house  valued  at  $6000  is  insured  for  f  of  its  value 
at  the  rate  of  J  of  1%   a  year.     How  much  is  the  annual 
premium? 

40.  Find  the  prime  factors  of  1226,  1938,  and  2346.     In- 
dicate which  of  these  factors  must  be  combined  to  produce  (a) 
the  greatest  common  divisor,  (b)  the  least  common  dividend. 


41.  Make  a  receipted  bill  of  the  following:  Sold  this  day 
to  Anson  White,  3  bbl.  flour,  at  $3.75  ;  75  Ib.  sugar,  at  5  cts. ; 
10  Ib.  coffee,  at  35  cts. ;  2  Ib.  tea,  at  60  cts. 

42.  Find  the  amount  at  simple  interest  of  $865.35  for  1  yr. 
5  mo.  17  da.  at  4J%. 

43.  In  a  certain  school  district  the  assessed  valuation  of 
property  is  $136,395,  and  the  amount  to  be  raised  by  local  tax 
is  $785.72.     Find  the  amount  of  A.'s  tax,  whose  property  is 
assessed  at  $8500. 

44.  A  bar  of  iron  in  the  form  of  a  cylinder,  6  feet  long  and 
2  inches  in  diameter,  is  forged  into  a  square  bar  whose  cross- 
section  is  2J  square  inches.     Find  the  length  of  the  new  bar. 

45.  A  man  plants  corn  on  £  of  his  land,  potatoes  on  2J 
times  as  much,  and  sows  the  remainder  with  wheat.     He  sells 
the  wheat  at  60  cts.  a  bushel,  and  receives  for  it  $180.    If  the 
yield  of  wheat  was  20  bushels  an  acre,  how  much  land  had  he  ? 


350  PRACTICAL  ARITHMETIC 

"46.  Simplify  the  following :  a8*  *  2V 

?    X    9    X    3 

47.  A  note  for  $624  is  dated  August  26,  1893;  July  15, 
1894,  there  was  paid  on  it  $62.50.    Find  the  amount  now  due. 

48.  Find  the  amount  of  $685  at  4J%  simple  interest  from 
July  1,  1894,  to  the  present  time. 

49.  Define  and  illustrate  dividend,  power,  ratio,  factor. 

50.  I  buy  hats  at  $18  a  dozen  and  sell  them  at  $2.50  apiece. 
Find  the  gain  per  cent. 


51.  I  sell  goods  at  a  discount  of  10%  from  the  marked 
price  and  still  make  a  profit  of  8%.     How  many  per  cent, 
above  cost  was  the  marked  price  ? 

52.  What  single  discount  is  equal  to  a  commercial  discount 
of  10%,  10%,  and  5%  ? 

53.  Find  the  square  root  of  1,080,234  to  two  decimal  places. 

54.  Find  the  least  possible  cost  of  carpeting  a  room  15  feet 
long,  12  feet  wide,  with  carpet  f  yd.  wide,  at  75  cts.  a  running 
yard. 

55.  Write  the  table  of  avoirdupois  weight.     For  what  is 
this  weight  used  ? 

56.  Two  men  start  from  the  same  point  on  a  level  plain  and 
travel,  one  due  north  at  the  rate  of  3  miles  an  hour,  the  other 
due  east  at  the  rate  of  5  miles  an  hour.     How  far  apart  will 
they  be  at  the  end  of  10  hours  ? 

57.  Divide  one  millionth   by  eight  ten-thousandths,  and 
express  the  result  in  words. 

58.  Find  the  prime  factors  of  2964,  and  all  the  different 
composite  factors  into  which  the  prime  factors  may  be  com- 
bined. 

59.  Define  minuend,  multiplication,  prime  factor,  common 
divisor,  ratio. 


GENERAL    REVIEW  351 

60.  Find  the  amount  at  simple  interest,  at  5%,  of  $860 
from  Sept.  1,  1894,  to  the  present  time. 


61.  Show  that  if  four  quantities  are  in  proportion  the  pro- 
duct of  the  means  equals  the  product  of  the  extremes. 

62.  How  much  is  due  Aug.  15,  1893,  on  an  interest-bearing 
promissory  note  for  $250,  dated  Buffalo,  June  1,  1886,  on 
which  $50  was  paid  Dec.  24,  1886,  and  $10  Jan.  5,  1888? 

63.  Find  the  cost,  at  $7  per  100  sq.  ft.,  of  slating  a  trape- 
zoid  of  which  the  parallel  sides  are  64  ft.  and  32  ft.,  and  the 
perpendicular  distance  between  them  is  20  ft. 

I  Ql      ^      >J2 

64.  Simplify  and  express  decimally  8J*_  61  • 

65.  Find  the  square  root  of  8.5849. 

66.  Find  the  cost  of  shingles  required  to  cover  a  roof  40 
ft.  long,  20  ft.  wide  at  $5.00  a  thousand,  if  it  requires  36 
shingles  to  cover  5  sq.  ft. 

67.  Find  the  amount  due  this  day  on  a  note  given  in  New 
York  May  10,  1890,  for  $500,  with  interest,  a  payment  of 
$35  having  been  made  July  5,  1891. 

68.  Reduce  to  its  lowest  terms  ,(1J  +  lf)4X  3.- 

-g  x  £-%  —  ^  —  f 

69.  Define  least  common  dividend,  factor,  numerator,  divisor, 
root,  proportion,  fraction. 

70.  A  cistern  is  6  ft.  square.     How  deep  must  it  be  to  hold 
30bbl.  of  water? 


71.  Find  the  least  common  dividend  (multiple)  and  the 
greatest  common  divisor  of  45,  70,  and  105. 

72.  How  many  times  will  a  wheel  4  ft.  in  diameter  revolve 
in  going  one  mile  ? 


352  PEACTICAL  ARITHMETIC 

73.  Find  the  diagonal  of  a  rectangle  whose  sides  are  15  ft. 
and  20  ft, 

74.  I  invest  $6000  in  6%   bonds  at  125.     What  rate  per 
cent,  do  I  receive  on  the  investment  and  what  is  the  income 
from  it  ? 

75.  A  field  is  42  rd.  long  and  35  rd.  wide.     Find  its  value 
ut  $37.50  an  acre. 

76.  A  man  6  ft.  high  casts  a  shadow  42  in.  long.     Find 
the  height  of  a  flagstaff  which  at  the  same  time  casts  a  shadow 
28  ft.  long. 

77.  Multiply  2  thousand  9  ten-millionths  by  30  thousand 
2  and  7  tenths,  and  divide  the  product  by  3  ten-thousandths. 

78.  An  agent  remits  to  me  $247.38,  after  retaining  a  com- 
mission  of  5%    for  collection.     What  sum  did  he  collect? 
What  was  the  amount  of  his  commission? 

79.  Three  men  engage  in  partnership.     A.  puts  in  $1200, 
B.  $1550,  C.  $1900.     They  gain  $350.     What  is  each  man's 
share  of  the  profits? 

80.  The  owner  of  -fj-  of  a  mine  sold  -f$  of  his  share  for 
$40,500.     What  should  he  who  owns  ^  of  the  mine  get  for  f 
of  his  share? 

81.  If  18  men  can  dig  128  yards  of  ditch  in  32  days,  how 
many  yards  can  1 2  men  dig  in  64  days  ? 

82.  If  a  square  field  contains  10  acres,  what  is  the  length 
of  the  diagonal  ? 

83.  At  what  price  must  6%  bonds  be  bought  to  yield  4% 
on  the  investment  ? 

84.  If  8  men  reap  36  acres  of  grain  in  9  days,  working  9 
hours  a  day,  how  many  men  will  reap  48  acres  in  12  days, 
working  12  hours  a  day? 

85.  Find  the  cost,  at  35  cts.  per  cubic  yard,  of  excavating  a 
trench  6  rods  long,  1  \  yards  wide,  1  foot  6  inches  deep. 


GENERAL    EEVIEW  353 

86.  A  note  for  $560,  payable  in  90  days,  is  discounted  at  a 
bank  30  days  after  it  is  dated.     Find  the  proceeds. 

87.  Find  the  amount  of  $945.15  from  December  15,  1891, 
to  November  22,  1892,  at  4J%  simple  interest. 

88.  Divide  $720  among  A.,  B.,  and  C.,  so  that  the  number 
of  dollars  they  receive  shall  be  as  the  numbers  5,  6,  and  7. 

89.  A  merchant  marks  an  article  $2.80,  but  in  selling  it 
takes  off  5%  for  cash.     If  the  rate  of  his  profit  is  33%,  what 
was  the  cost  of  the  article  ? 

90.  What  part  of  an  ounce  is  53  %3  ? 


91.  Find  the  amount  of  $375  for  11  mo.  17  da.,  at 
simple  interest. 

92.  Find  the  cost,  at  25  cts.  a  rod,  of  building  a  fence  round 
a  square  10-acre  field. 

93.  How  many  gold  rings,  each  weighing  5  pwt.  18  gr., 
can  be  made  from  2  oz.  6  pwt.  of  gold  ? 

94.  Find  the  face  of  a  60-day  note  which,  when  discounted 
at  a  New  York  bank,  will  yield  $250. 

95.  If  it  costs  $80  to  plough  a  field  40  rods  by  80  rods 
when  we  pay  $5  a  day  for  man  and  team,  how  much  will  it 
cost  to  plough  a  field  30  rods  by  60  rods  if  we  pay  $4  a  day  ? 

Suggestion  :  Solve  by  proportion  and  by  analysis. 

96.  What  number  divided  by  the  sum  of  £  and  2^  will 
give  a  quotient  of  2^-  ? 

97.  If  rain-drops  are  falling  directly  downward,  how  much 
more  ground  surface  would  be  protected  from  the  rain  by  a 
board  20  feet  long  and  18  inches  wide  when  in  a  horizontal 
position  than  when  one  end  of  it  is  elevated  9  feet  higher  than 
the  other? 

23 


354  PRACTICAL   ARITHMETIC 

98.  A  certain  town  raised  a  tax  of  $4607.50.     The  real 
estate   was   valued   at   $420,000,    the   personal   property   at 
$189,000,  and  1250  persons  paid  a  poll-tax  of  $1.25  each. 
Find  the  tax  on  $1.00  of  the  property. 

99.  How  high  must  be  a  pile  of  wood  10  feet  long  and  2J 
feet  wide  to  contain  one  cord  ? 

100.  How  much  should  be  paid  for  40  shares  of  railroad 
stock  at  3J%  discount  and  \<J0  brokerage? 


101.  Find  the  diagonal  of  a  cubical  block  each  of  whose 
edges  is  20  inches. 

102.  How  many  dollars  would  a  man  gain  in  buying  240 
shares  of  railroad  stock  at  3f  %  discount  and  selling  them  at 
1|%  premium? 

103.  A  note  for  $350,  dated  October  17,  1865,  was  paid 
April  11,  1868,  with  interest  at  7%.     Find  the  amount  paid. 

104.  What  would  be  the  cost  of  50  boards,  each  12  feet 
long,  8  inches  wide,  and  1J  inches  thick,  at  4J  cts.  a  foot, 
board  measure? 

105.  At  30  cts.  a  sq.  yd.,  IIOAV  much  will  it  cost  to  plaster 
the  four  walls  and  ceiling  of  a  room  15  ft.  X  18  ft.  and  9  ft. 
high,  no  allowance  being  made  for  openings? 

106.  For  what  sum  must  I  make  a  bank  note  at  90  days, 
that  the  proceeds  may  be  $150? 

107.  Simplify  8|  +  ^-4t. 

108.  Find  the  true  discount  and  the  present  worth  of  $412, 
due  in  6  mo.,  without  interest. 

109.  Find  the  cost  of  1478  Ib.  of  coal  at  $4.60  per  ton. 

110.  How  much  lumber  will  be  required  to  ceil  the  four 
walls  of  a  room  16  ft.  X  18  ft,  and  10  ft.  high,  and  how 
much  will  the  lumber  cost  at  $16  per  M,  ? 


GENERAL    REVIEW  355 

111.  If  5  men  can  dig  a  trench  10  rods  long,  2  ft.  wide, 
and  5  ft.  deep  in  4  days,  how  many  men  will  it  take  to  dig  a 
trench  40  rods  long,  2  ft.  wide,  and  4  ft.  deep  in  8  days  ? 

Suggestion  :  Solve  by  proportion. 

112.  I  buy  stocks  at  80  and  sell  them  at  par.     Find  the 
per  cent,  profit. 

113.  A.  owns  ^  of  a  farm  worth  $15,422,  and  sells  f  of 
his  share.     Find  the  value  of  what  he  has  left. 

114.  A  pension  of  $140  per  year  is  four  years  in  arrears. 
Find  the  amount  now  due  at  5%  compound  interest. 

115.  How  many  bushels  will  a  bin  contain  that  is  9  ft. 
long,  4  ft.  wide,  6   ft.  deep?     How  many  bushels,  heaped 
measure  ? 

116.  Constantinople  is  in  longitude  28°  59'  E.  and  Phila- 
delphia 75°  10'  W.     When  it  is  4  A.M.  in  Philadelphia,  what 
time  is  it  at  Constantinople? 

117.  Find  the  diagonal  of  a  right  parallelepiped  whose 
edges  are  6  ft.,  8  ft.,  and  4  ft. 

118.  Find  in  inches  to  two  places  of  decimals  the  diagonal 
of  a  cube  whose  volume  is  9  cu.  ft. 

119.  The  diameter  of  the  base  of  a  cone  is  double  that  of 
the  base  of  a  cylinder  of  the  same  volume.     Find  the  ratio 
of  their  altitudes. 

120.  A  locomotive  runs  f  of  a  mile  in  £  of  a  minute.    How 
many  feet  does  it  run  in  a  second  ? 


121.  The  base  of  a  certain  triangle  is  40  ft.,  its  altitude  is 
30  ft.     Find  the  area  of  a  similar  triangle  whose  base  is  25  ft. 

122.  I  buy  an  article  by  avoirdupois  weight  and  sell  it  at 
the  same  price  per  pound  by  Troy  weight.     Do  I  gain  or  lose, 
and  how  many  per  cent.  ? 


356  PRACTICAL    ARITHMETIC 

123.  Find  the  cost  of  a  draft  on  Chicago  for  $1000  at  60 
days'  sight,  money  being  worth  5%  and  exchange  at  1J% 
premium. 

124.  When  it  is  noon  in  Philadelphia,  what  is  the  time  in 
Paris,  2°  20'  E.  long.?     In  San  Francisco,  122°  25'  40.76" 
W.  long.  ? 

125.  A.  is  in  longitude  18°  E.  and  B.  23°  W.     Find  the 
difference  of  time  between  A.  and  B.,  and  give  the  reason  for 
each  step  in  the  process. 

126.  Find  the  depth  of  a  cylindric  cistern  whose  bases  are 
8  ft.  in  diameter  and  whose  capacity  is  100  barrels. 

127.  Find  the  amount  of  $436   at  4J%   simple  interest 
from  January  1,  1893,  to  the  present  time. 

128.  The  wheels  of  a  sulky  are  4J  ft.  apart.     In  driving 
around  a  circular  track,  the  inner  wheel   traverses   1   mile. 
How  far  does  the  outer  wheel  go  ? 

129.  Write  a  full  analysis  of  the  following  :  f  is  £  per  cent, 
of  how  many  times  -f  ? 

130.  Find  the  face  of  a  note  at  60  days,  without  interest, 
which  will  yield  $750  proceeds  when  discounted  at  a  New 
York  bank. 

131.  Find  the  exact  interest  of  $590  from  Sept.  18,  1893, 
to  March  1,  1894,  at  4J%. 

132.  Find  the  side  of  a  square  which  is  equal  in  area  to  a 
right  triangle  Avhose  base  is  24  ft.  and  hypotenuse  40  ft. 

133.  Find  the  cube  root  of  1796.63  to  two  places  of  deci- 
mals. 

134.  Find    the   surface   and   volume   of  a   sphere   whose 
diameter  is  4  ft. 

135.  Find  the  volume  and  the  entire  surface  of  a  square 
pyramid  the  side  of  whose  base  is  2  ft.,  and  whose  slant 
height  is  6  ft. 


GENERAL    REVIEW  357 

136.  Find  the  smallest  number  that  will  exactly  contain 
15,  18,  21,  24,  30,  and  91. 

137.  Find  the  rate  per  cent,  of  interest  on  an  investment 
in  government  3%  bonds  bought  at  115. 

138.  Find  the  value  of  43,562  X  21,894  +  986. 

139.  Show  your  knowledge  of  the  use  of  signs  by  indicat- 
ing the  solution  of  the  following :   A  man  earns  $37.50  a 
month  for  6    months   and  $50  a  month  for  9  months.     He 
invests  his  earnings  in  railway  stock  at  75.     The  stock  pays 
a  dividend  of  4%,  and  the  money  thus  received  is  divided 
among  his  children,  each  receiving  as  many  dollars  as  there 
are  children.     How  many  children  are  there  ? 

140.  If  it  requires  40  min.  for  a  pipe  4  in.  in  diameter  to 
fill  a  tank  20  ft.  by  10  ft.  by  6  ft.,  how  long  will  it  take  a  pipe 
3  in.  in  diameter  to  fill  a  tank  30  ft.  by  12  ft.  by  8  ft. 


141.  If  a  ball  2|  ft.  in  diameter  weighs  400  lb.,  what  is 
the  diameter  of  a  similar  ball  that  weighs  1  T.  1200  lb.  ? 

142.  A  box  made  of  2-inch  plank  is  3  feet  4  inches  long1, 
2  feet  8  inches  wide,  and  1  foot  6  inches  high  ;  it  has  no  lid. 
How  much  will  it  cost  to  cover  the  box  completely  inside  and 
outside  with  gold  leaf,  at  $2  per  square  foot? 

143.  A  bushel  measure  and  a  peck  measure  have  been  made 
of  the  same  shape.     Find  the  ratio  of  their  heights. 

144.  How  many  feet  of  lumber  are  there  in  three  1  J-inch 
16-foot  boards  whose  breadths  are  respectively  12,  14,  and  15 
inches  ? 

145.  A  railway  one  chain  wide  runs  across  a  section  of  land 
parallel  with  one  side.     Find  the  price  of  the  land  in  the  rail- 
way at  $25  per  acre  ? 


358  PKACTICAL    ARITHMETIC 

146.  If  the  ox  that  Milo  carried  was  6^  ft.  in  girth  when 
it  weighed  1000  lb.,  what  was  the  girth  of  the  ox  when  it 
weighed  2000  Ibs.  ? 

147.  If  I  pay  $ .62^-  a  cord  for  sawing  wood  4  feet  long  into 
3  pieces,  how  much  more  should  I  pay  for  sawing  wood  8  feet 
long  into  pieces  of  the  same  length  ? 

148.  A  room  is  18  ft.  long,  15  ft.  wide,  and  10  ft.  high. 
What  is  the  distance  from  an  upper  corner  to  the  opposite 
lower  corner  ? 

149.  If  a  5-in.  cube  of  granite  weighs  12  lb.,  what  will  a 
cubic  foot  weigh  ? 

150.  I  have  a  trapezium  of  land,  measuring  30,  40,  60,  70 
rd.,  with  a  diagonal  of  50  rd.     Find  the  area  of  the  field. 

151.  Given  the  frustum  of  a  square  pyramid :  height,  20 
ft.  ;  side  of  upper  base,  8  in. ;  side  of  lower  base,  20  in. 
Find  its  volume. 

152.  Find  the  surface  of  a  cube  that  contains  5268.024 
cubic  inches. 

153.  A  spherical  balloon  contains  28974.25  cubic  feet. 
Find  the  number  of  square  yards  of  silk  required  to  make  it. 

154.  What  must  be  the  market  price  of  3%  stock,  that  it 
may  give  ?>\%  interest  after  deducting  35  cts.  from  every  $12 
of  the  income  ? 

155.  David  Palmer  borrows  this  day  of  Samuel  Hill  $350, 
and  gives  his  note  for  this  amount  for  4  months  at  6%. 
Make  out  the  promissory  note  in  proper  form. 

156.  When  it  is  3  P.M.  at  Rome,  longitude  12°  27'  east, 
it  is  8.20  A.M.  at  Chicago  ;  find  the  longitude  of  Chicago. 

157.  A  and  B  run  a  mile  in  opposite  directions :  A's  run- 
ning is  to  B's  as  6-J- :  5£  ;    B  gets  4  seconds  start,  during 
which  time  he  runs  12^  yards.    Find  when  he  will  pass  A. 


ANSWERS. 


Page  14. 

1.1.  Fifteen. 

2.  Four. 

3.  Fourteen. 

4.  Twenty-four. 

5.  Nineteen. 

6.  Thirty-nine. 

7.  Thirty-three. 

8.  Twenty-nine. 

9.  Forty-nine. 

10.  Forty-five. 

11.  Ninety -nine. 

12.  Sixty-five. 

13.  One  hundred  nine. 

14.  One  hundred  eleven. 

15.  Ninety-one. 

16.  Six  hundred  ninety. 

17.  Three  hundred  39. 

18.  Seven  hundred  34. 

19.  790. 

20.  1029. 

21.  5555. 

22.  550600. 

23.  210506. 

24.  8000. 

25.  200090. 

26.  149. 

27.  2500. 

28.  70899. 

29.  1595864. 


Page  15. 

2.  1.  XV. 

2.  XXXVI. 

3.  LXXXVII. 

4.  LVI. 

5.  XLIX. 

6.  XCIX. 

7.  ML. 

8.  MMMMMX  =  VX. 

9.  DCCLXXXIX. 

10.  MDCCCXCVIII. 

11.  XVIII. 

12.  XLII. 

13.  LXVI. 

14.  LXXXVI. 

15.  LXIII. 

16.  C. 

17.  IIIDC.  or  MMMDO. 

18.  DLXXXVII. 

19.  CCVII. 

20.  VIIIIV. 

21.  XXVII. 

22.  LXXXI. 

23.  XCV. 

24.  XL. 

25.  XLV. 

26.  DXXXIV. 

27.  V. 

28.  CDXXXVI. 

29.  CMXCIX. 

30.  LXXVTCMLIX, 

359 

360 


ANSWERS 


ADDITION. 

Page  23. 

Page  18. 

21.  1.  66892.            5.  67573. 

1.  1.  599.                   6.  $6.95. 

2.  58434.            6.  46997. 

2.  676.                   7.  $9.55. 

3.  508785.          7.  51871. 

3.  1026.                8.  $92.79. 

4.  $9622942.     8.  49845. 

4.  794.                  9.  $983.9Q 
5.  748. 

Page  24. 

Page  19. 

9.  134083.44.     11.  2356.9657. 

10.  16193.            12.  10615. 

10.  108349.98. 

11.  10333             13,  7720. 

2.  8649. 

SUBTRACTION. 

3.  1.  11429.              4.  77230. 

Page  26. 

2.  20681.              5.  235308. 

1.  411.             3.  254.             5    352. 

3.   101391. 

2.  324.            4.  213.            6.  5533 

4.  1.  4164.                1.  2519. 

2.  1461.                2.  3046. 

Page  27. 

3.  2867.                3.  1965. 

7.  $25.62.                11.  32,154. 

4.  3285.               4.  2690. 

8.  $35.09.                12.  27,312. 

5.  2791.                5.  3332. 

9.  $11.13.                13.  422.641. 

6.  1453.                6.  2469. 

10.  $21.40.                14.  145.325. 

5.  104367.                 7.  83619. 

2.  1520. 

6.  $1447.845.            8.  $132.90. 

Page  28. 

Page  2O. 

4.  $3264.                      8.  127,420. 
5.  1212.                        9.  289. 

9.  7076.                     10.  31164. 

6.  6,550,216.              10.  9  yrs. 

2.  95  acres.                 4.  20694. 

7.  $4,820,411. 

3.  $604.20.                  5.  1065. 

Page  SO. 

Page  21. 

1.  1.  305.                  12.  2131. 

6.  528408.                 10.  402399. 

2.  228.                  13.  $3.07. 

7.  13587.                   11.  $2454.63. 

3.  292.                  14.  $2.17. 

8.  $2275.00.              12.  $8513.75. 

4.  272.                  15.  $16.17. 

9.  58639. 

5.  1879.                16.  $24.96. 

Page  22. 

6.  61.                    17.  $33.66. 

13.   1646619.              18.  61. 

7.  1919.                18.  $.995. 

14.  73941.                 19.  LXII. 

8.  388.                  19.  $88.996. 

15.  365.                      20.  $1906.50, 

9.  1299.                20.  1,410,273. 

16.  72  days.                      $6140.66, 

10.  40.                   21.  3120473. 

17.  $171800.                   $8047.16. 

11.  6828.               22.  998.78. 

ANSWERS 


361 


2.  1.  27,747.             6.  132,890. 

MULTIPLICATION. 

2.  45,860.            d.  430,875. 
3.  493,879.           7.  5,741,182. 

Page  39. 

4.  382,717.           8.  1,987,588. 

2.  1.  730.                    6.  $25.45. 

2.  2696.                  7.  $63.60. 

Page  31. 

3.  2268.                  8.  $89.25. 

4.   1962.                  9.  $495  54. 

2.  $2125.                     8.  7795. 

5.  2040.                10.  $523.20. 

4.  45,558,897.             9.  69,191,517. 
5.  45yrs.                    10.  $3149. 
6.  67yrs.                    11.  $925,985. 
7.  7600  ft. 

4.  1.  69536.                6.  41C88. 
2.  37296.                7.  261045. 
3.  51590.                8.  478709. 

4.  65601.                9.  318352. 

Page  32. 

5.  69380.              10.  827847. 

12.  $1046.                  17.  4908  ft. 

5.  1.  3780.                  5.  162108. 

13.  $4.365.                18.  1437. 

2.  18118.                6.  1526190. 

14.  $19.81.                19.  1706. 

3.  234177.               7.  243582. 

15.  $27,404.              20.  LXlVII. 

4.  12533346.           8.  7282896 

16.  14,162ft. 

6.  1.  302.         4.  687.         7.  2484. 

1.  23,527.              2.  33,958. 

2.  54.          5.  8537. 

3.  4160.       6.  1553. 

Page  33. 

3.  13,181.                 5.  $207.61. 

Page  4O. 

4    120,091. 

2.  $11.25.                    7.  672. 

1.  224,980                3.  919. 

3.  1227.45.                 8.  2,400,000. 

2.  19,553068.          4.   55. 

4.  15840.                     9.  82287UOOO. 

5.  $29316.                 10.  $2499.96. 

Page  34. 

6.  $6655.                   11.  $42,592. 

2,  165.              6.  91,145;  58,905. 

8.  $1365.          7.  3561. 

Page  41. 

4.  1155.            8.  $4484. 

12.  .03.                      14.  Lost  $10. 

5.  2070.            9.  D.  697. 

13.  Cows,  20.            15.  9050. 

Page  35. 

Page  42. 

10.  447.                     15.  796. 

2.  1.  4755.                6.  $440.55. 

11.  245.                     16.  96. 

2    7728.               7.  $767.55. 

12.  115.                     17.  168.89. 

3.  19481.              8.  $2176.56. 

13.  273.                     18.  82. 

4.  17082.              9.  $3477.33. 

14.  2410.                   19.  11,220. 

6.   12691.            10.  $1614.14. 

362 


ANSWERS 


3.  1.  7258.     11.  59424. 

Page  45 

2.  13440.    12.  66822. 

1.  4860,  48600,  194400. 

3.  21465.    13.  47320. 
4.  19758.    14.  45384. 

2.  382400,  764800,  9560000. 
3.  1722000,  2296000,  2583000. 

5.  47085.    15.  78027. 
6.  45522.    16.  21909. 

4.  747000,  7470000,  10956000. 
5.  21492000,256710000,453720000. 

7.  42182.    17.  88445. 
8.  66822.    18.  90159. 

6.  13536000,  156510000, 
2411100000. 

9.  53963.    19.  229554. 

7.  1315170000000,  1480784000000. 

10.  47974.    20.  307395. 

8.  $3139972.00,  $39249650.00. 

4.  31806. 

9.  3604200000,  2405202800000. 

5.  138104.50. 

10.  440000000,  25,960,000. 

6.  119239. 

12.  3168000. 

7.  350090. 

13.  126000000. 

8.  46529640. 

14.  48000. 

9.  $16808.61. 

15.  1610000. 

10.  1.  $6141.720.  3.  27154202. 

16.  1140000. 

2.  5107212.   4.  96332187. 

17.  44000000. 

Page  43. 

Page  46. 

11.  35843685. 

12.  214007086881. 

18.  $650.        20.  86,400. 

1  Q  $80  000 

13.  764,819,895,290,424. 

At7»  ipOvjUV/v. 

14.  2,324,334,767,296. 

15.  99253.80. 

Page  47. 

16.  152323.35. 

1.  333641.       5.  3306564. 

17.  $69520.33. 

2.  27421443.      6.  401193. 

18.  CMXII. 

3.  14889792.     7.  2153232. 

19.  CLXXXVIICDLVI. 

4.  4382415.      8.  49308. 

20.  5859385041295896. 

21.  21,842,100. 

1.  500.     3.  $72.     5.  9328. 

2.  $503.50.  4.  913,920. 

1.  $20604.       4.  984072. 

2.  11025.        5.  $494. 
',}  $2533  50.      6.  137664. 

Page  48. 

6.  320,000. 

Page  44. 

7.  213,192. 

7.  95040. 

8.  $1,377. 

8.  (65  —  57)  X  54  ==  432. 

9.  103,615 

9.  (17  +  2fi)  X  $42.50  —  (17  X 

10.  $2,583. 

38.75)  -f  (26  X  40.25)  = 

11.  89,232 

$122.25. 

12.  $31.80. 

ANSWERS 


363 


13.  5,865,696,000,000. 

17.  93716.       22.  $.92. 

14.  13,176. 

18.  209758.      23.  $108.50. 

15.  968710. 

19.  189572.      24.  32793. 

16.  $17,979,365. 

20.  26485.       25.  11750. 

21.  $6.07.        26.  63362. 

Page  49. 

2.  406.         3.  432. 

17.  276. 

18.  693. 

Page  56. 

19.  63,210,541,205,000. 

4.  167.           13.  32. 

1.  (16  —  11  -f  2)  X  6  =  35. 

5.  172,  with4rem.   14.  123. 

2.  (4-f  15)X(15  —  4)X  6  =  1254. 

6.  230,  with  7  rem.   15.  182. 

3.  63915  +  936085  =  1000000. 

7.  403.           16.  92. 

4.  3149  -f-  4872  =  8021. 

8.  286,  with  8  rem.   17.  88. 

5.  5301  _  1046  =  4255. 

9.  315,  with  14  rem.  18.  217. 

6.  300,003,  300,003. 

10.  312.           19.  136. 

7.  MMMIII,  CI. 

11.  1899,  with  5  rem.  20.  72.' 

8.  397,056. 

12.  439,  with  38.  rem.  21.  35. 

9.  12,343,200. 

2.  95.    4.  144.    6.  108. 

10.  674. 

3.  8.     6.  13178. 

DIVISION. 

Page  57. 

Page  54. 

7.  81.    15.  365. 

1.  283.    9.  5263.   17.  209758. 

8.  104.   16.  5280. 

2.  188.   10.  6238.   18.  189572. 

9.  37.    17.  17443,  with  16  rem. 

3.  71.    11.  4812.   19.  $264.85. 

10.  72.    18.  175. 

4.  124.   12.  4809.   20.  $924.67 

11.  66.    19.  327. 

5.  834.   13.  247.    21.  $128.21. 

12.  162.   20.  7328. 

6.  4169.  14.  4138.   22.  $222.22. 

13.  2640.  21.  $89. 

7.  9451.  15.  6559. 

14.  153. 

8.  9485.  16.  93716. 

Page  58. 

1.  283.     4.  248.     7.  9451 

1.  1.  11,572,  with  110  rem. 

2.  198.     5.  1668.    8.  9485. 

2.  6284. 

3.  142.     6.  4169. 

3.  1938. 

4.  664. 

5.  736. 

Page  55. 

6.  893. 

9.  5263.        13.  27680. 

7.  969,  with  344  rem. 

10.  6238.        14.  2470. 

8.  1064. 

11.  4812.        15.  8276. 

9.  985. 

12.  4809.        16.  6559. 

10.  692,  with  533  rem. 

364 


ANSWERS 


3.   1.  527,  with  380  rem. 

1.  576,544.          9.  319,099. 

2.  692,  with  533  rem. 

2.  103,075.        10.  801,587. 

3.  5205,  with  38  rem. 

3.  213,789.        11.  117,554. 

4.  814,  with  167  rem. 

4.  4.                   12.  388,129. 

5.  1259,  with  581  rem. 

5.  9,042,049.     13.  8,886,859. 

(;.  645,  with  312  rem. 

6.  8161.             14.  253,  with  21,  700 

7.  283,  with  736  rem. 

7.  1162.                        rem. 

8.  3241. 

8.  28. 

99401 

OTzOX. 

10.  876,  with  110  rem. 

Page  62. 

11.  474,536,  with  523  rem. 

15.  17,115,520. 

12.  4567. 

16.  . 

13.  4207. 

17.  67,  with  999  rem. 

14.  10,110,  with  9  rem. 

18.  25. 

19.  240. 

4.  1.   1,672,940,  with  165,534  rem. 
2.  206,008,604,  with  24  rem. 

20.  300,  with  9999  rem. 

3.  100,000,000,  with  102,345,678 

2.  18.               3.  276.            4.  40. 

rem. 

4.  100,000,000. 

Page  63. 

5.  48,100,720,009. 

5.  $10.00. 

6.  3000  too  much  in  2d  member. 

Page  59. 

7.  41. 
8.  132699. 

1.  96.             6.  329.            10.  58. 

9.  90. 

2.  1760.         7.  85  +.         11.  7  +. 

10.   2122. 

3.  60.             8.  22  -}-.         12.  3579. 

11.  19,868. 

4    19.            9.  36.               13.  25. 

12.  139,806. 

5.  425. 

13.  $384.25. 

Page  6O. 

14.  Lost  $952. 

H.  491  sec.               21.  548,501. 

15.  5475  hr. 

15.  15.                       22.  9238. 

16.  12,295. 

16.  500.                     23.  5. 

Page  64. 

17.  605.                     24.  28. 
18.  $137  nearly.       25.  308. 
19.  357  -f.                26.  0. 

17.  5  yr.                    19.  2.83. 
18.  105.                     20.  $12.40. 

20.  13.                       27.  IX. 

Page  65. 

Page  61. 

2.  $3.60.       5.  $.66.            8.  $2552 

28.  CLXXX.           30.  437. 

3.  $6.10.       6.  $4.50.          9.   116. 

29.  MCCLXXX 

4.  $3125.      7.  $28.00.      10.  6480. 

ANSWERS 


365 


Page 

66. 

PROPERTIES    OF    NUM 

2. 

93. 

4.  12 

5.  47. 

BERS. 

1. 

20. 

3.  1785. 

5.  5. 

Page  74. 

2. 

57, 

656. 

4.  210. 

1. 

9 

—  ) 

2, 

2O       O 
,  6,  6. 

2. 

5 

7. 

Page 

67. 

3. 

2, 

2, 

2,  2,  2, 

2. 

7 

36. 

11.  10 

14. 

$60. 

4. 

A 

23 

8 

10  hr. 

12.  $.25. 

15. 

$3010. 

5. 

2, 

2, 

2,  2,  3, 

7. 

9 

114. 

13.  $1 

50. 

16. 

40. 

6. 

3, 

37 

10 

108. 

7. 

5 

7, 

11. 

Page 

69. 

8. 

3, 

11 

,  13. 

Q. 

Rem. 

Q- 

Rem. 

9. 

5, 

5, 

37. 

2. 

1. 

632, 

7. 

9. 

55, 

33. 

10. 

2, 

2, 

3,41. 

2. 

532, 

7. 

10. 

12, 

34. 

11. 

2, 

2, 

2,  3,  5, 

11. 

3. 

973, 

2. 

11. 

6, 

173. 

12. 

2, 

2, 

2,  3,  3, 

3,  3,  18. 

4. 

926, 

7. 

12. 

5, 

432. 

13. 

2, 

3, 

1283. 

5. 

256, 

7. 

13. 

8, 

650. 

14 

743, 

prime  number. 

6. 

32, 

67. 

14. 

3, 

000. 

15. 

3, 

5, 

5,  7,  7. 

7. 

53, 

27. 

15. 

5, 

678 

16. 

2, 

2, 

2,  3,  3, 

3,  3,  7. 

8. 

92, 

73. 

17. 

2, 

2, 

2,  3,  7. 

Q. 

Rem. 

Q. 

Rem. 

18. 

3, 

7, 

11. 

2. 

1 

33, 

13. 

5. 

13, 

34. 

19. 

2, 

89. 

2. 

31, 

27. 

6. 

12, 

16. 

20. 

2, 

2, 

3,  3,  5. 

3. 

17, 

6. 

7. 

28, 

136. 

21. 

2, 

2, 

2,  2,  3, 

3. 

4. 

15, 

40. 

8. 

24, 

100. 

22. 

3, 

3, 

5,7. 

23. 

2, 

2, 

3,  5,  7. 

Page 

70. 

24. 

2, 

2, 

3,  5,  11. 

Q. 

Rem. 

Q. 

Rem. 

25. 

2, 

2, 

5,37. 

3. 

1. 

1, 

273 

7. 

2, 

2432. 

26. 

3, 

3, 

3,  5,  7. 

2. 

1, 

3o2. 

8. 

2, 

37. 

27. 

2, 

2, 

2,  2,  2, 

2,  3,  3,  3. 

3. 

1, 

295 

9. 

1, 

3396. 

28. 

2, 

2, 

3,  3,  7, 

17. 

4. 

1, 

173. 

10. 

1, 

2116. 

29. 

2, 

29,  29. 

5. 

2, 

1327. 

11. 

1, 

2370. 

30. 

997, 

prime  number. 

6 

2, 

2645. 

12. 

1, 

1573. 

31. 

2, 

2, 

3,  5,  7, 

11. 

Q. 

Rem. 

Q. 

Rein. 

32. 

2, 

3, 

5,  5,  5, 

7. 

2. 

1. 

29, 

1958. 

7. 

45, 

5896. 

33. 

2, 

3, 

7,19. 

2. 

12, 

4425. 

8. 

20, 

17432. 

34. 

2, 

2, 

11,  11. 

3. 

14, 

4495. 

9. 

10, 

1959. 

35. 

2, 

2, 

2,  2,  2, 

2,  2,  2,  5. 

4. 

20, 

1765. 

10. 

38, 

9938. 

36. 

2, 

13,  73. 

5. 

10, 

4543 

11. 

23, 

25548. 

37. 

2, 

2, 

3,  6,  7, 

13. 

6. 

2725,  250. 

12. 

14, 

1337. 

38. 

2 

2, 

3,  3,  5, 

19. 

366 


ANSWERS 


39.  2,  3,  5,  7,  7. 

2.  1.  563.                   7.  89. 

40.  2,  2,  373. 

2.  324.                    8.  16. 

41.  2,  3,  5,  7,  11. 

3.  728.                    9.  38. 

42.  2,  2,  3,  3,  7,  11. 

4.  18.                    10.  864. 

43.  2,  2,  2,  2,  2,  2,  5,  5. 

5.  53.                    11.  892. 

44.  2,  2,  3,  17,  41. 

6.  430.                 12.  2735. 

45.  6,  11,  47. 

46.  1997,  prime  number. 

Page  77. 

47.  3,  3,  7,  7,  11. 

48.  3,  3,  7,  11,  11. 

Q.       R.                         Q.     R. 
1.  304,    8.               5.     44,  28. 

Page  75. 

2.     91,     9.                6.  119,  16. 

1 

3.  121,     9.                7.     49,  20. 

{2  X    3  =  6. 

2  X  17  =  34. 

4.     58,  27.               8.     23,  11. 

3X17  =  51. 

r  3  X  6  15 

Page  78. 

105.  |  3  X  7  =  21. 

3.  1.  45.      3.  Y-       5.  46.     7.  20. 

U  X  7  =  35. 

2.  45.      4.  21.        6.  4. 

2x2  =  4. 

2X3  =  6. 

2.  39. 
Page  79. 

108  ^  2  X  3  X  3  =  18. 
j  2  X  3  X  3  X  3  =  54. 

3.  20.                            9.  20. 

1  2  X  2  X  3  =  12. 

4.  1.50.                       10.  9. 

12x2x3x3  =  36. 

5.  2745.                      11.  8. 

221  has  prime  factors  only. 

6.  43.  80  very  nearly.  12.  21. 

r   5  X  U  =  55. 

7.  6.                             13.  26. 

715.  |    5X13  =  65. 

8.  6. 

Ill  X  13  =  143. 

Page  81. 

845  I    5  X  13  =  65' 
118X  13  =  169. 

2.   1.  7.        3.  7.       6.  6.       7.  6. 

The  answers  to   the  remaining 

2.  6.        4.  9-        6.  6.       8.  35. 

eleven  examples  are  omitted. 

3.  1.  14.      3.  15.      5.  16.     7.  120. 

2.     1.  $1855.            3.  $334.80. 

2.  42.      4.  20.      6.  42.     8.  22. 

2.  $227.70. 

Page  76. 

Page  82. 

4.  $215.82.         10.  $67.155. 

1.  9.  3.        '     15.  13.           21.  3. 

5.  694.95.           11.  $17424. 

10.  12.           16.  14.          22.  37. 

6.  $6375.            12.  $12600. 

11.  9.              17.  60.          23.  101. 

7.  $61.25.           13.  $87.04. 

12.  6.             18.  72.          24.  2. 

8.  $118.125.       14.  $76.95 

13.  75.            19.  29. 

9.  $1165.50. 

14.  144.         20.  1. 

ANSWERS 


367 


2.  1.  11.            8.  37. 

15.  37. 

7.  4284. 

2.  23.            9.  283. 

16.  47. 

8.  160,121. 

3.  31.           10.  2. 

17.  41. 

9.  441.000. 

4.  41.           11.  3. 

18.  53. 

10.  7770. 

5.  47.           12.  17. 

19.  267. 

11.  290,177. 

6.  53.           13.  48. 

20.  396. 

12.  1,639,872. 

7.  61.           14.  11. 

13.  314,259. 

14.  86,394. 

' 

Page  83. 

15.  1,009,091. 

3.  1.  12.            5.  43. 

8.  126. 

16.  1,038,007. 

2.  8.              6.  1. 

9.  42. 

17.  240,463. 

3.  4.              7.  3. 

10.  37. 

18.  179,655. 

4.  15. 

19.  50,552. 

1.  4.                  4.  12. 
2.  6.                  5.  14. 

3f»q 

6.  16. 

7.  23. 

20.  473,989. 
21.  23,760. 
22.  71,842,008. 

*    DO. 

Page  84. 

23.  31,154,994,649. 

8.  2.                 9.  5. 

10.  940. 

24.  260,117. 
25.  329,616. 

26.  4340. 

• 

Page  86. 

27.  42,149,000. 

1.  84.            7.  1080. 

13.  330. 

28.  3,268,080. 

2.  720.          8.  840. 

14.  720. 

3.  448.          9.  1200. 
4.  144.        10.  1440. 

15.  1200. 
16.  225. 

Page  89. 

5.  180.         11.  2016. 

17.  576. 

1.  840.      5.  210.       9. 

720. 

6.  360.         12.  360. 

18.  900. 

2.  180.      6.  720.      10. 

876. 

3.  5040.     7.  60.        11. 

82,063,340. 

Page  87. 

4.  120.       8.  460. 

1.  300.        7.  $10,800. 

12.  216. 

2.  120.        8.  630. 
3.  280.        9.  156. 

13.  5040. 
14.  510. 

Page  9O. 

4.  54.         10.  72. 

15.  10,920. 

1.  490.        5.  3. 

8.  509. 

5.  540.       11.  72. 

16.  6300. 

2.  3.            6.  40,170. 

9.  $10. 

6.  1512. 

3.  283.        7.  119. 

10.  720. 

Page  88. 

4.  1044. 

1.  2871. 

2.  13,889. 

FRACTIONS. 

3.  10013. 

4.  819. 

Page  94. 

5.  2160. 

2.  1.  ff.            3.  ft. 

6-  Jft. 

6.  2873. 

2.  ft-        4.yflh,. 

6-Tttr- 

368 


ANSWERS 


Page  95. 

Page  98. 

7. 

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19.  i§. 

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7.  £ 

Mf, 

if. 

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lili 

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ANSWERS 


369 


17.  W,  W,  H,  if 

11-  1-  if       5.  T^L          9.  1211. 

is.  f§,  -I,0,  M,  *g,  if 

2-  yVV     6-  ^32--         10.   18ff. 

19-  ihr°o>  IPf,  TjVi&i  tfM,  l§fj> 

3.  i.         7.  1048if.   11.  12|i. 

MM- 

4.  ft.       8.  52111      12.  41TW 

OA        7       125      18      140      12      30 

2  1  .    -3-6  2-$  8-,  -3-5-1-1  «-,  -2-8  -8-°  -°-  ,  JL3L&A.QL 

Page  1O6. 

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22.   j3^,   AW,   roV^J  TeVV,   f§il» 

2.  29ft.     9.  652T.        16.  15if^. 

iVsV 

3.  4f,       10.  17.          17.  15. 

4.  Oft.     11.  89f.        18.  If 

ADDITION. 

5.  261.     12.  45|.        19.   llff 

Page  102. 

1.  2||.         5.  lif.          9.  lOff 
2.  17ft.        6.  84ft.      10.  Uf- 
3.  31.            7.  12ft.      11.  26H- 

6.  79ft.  13.  14if,      20.  27f^. 
7.  9.         14.  34f|5.    21.  35ff. 
13.  1.  If.         6.  10^.      11.  4ff. 
2.  f.           7.  f^.        12.  1251. 

3Q77             Q      O  3  0  1           1Q      QQ23 

4.  4ii|.        8.  lOJff    12-  2H- 

T8^*                       ^^^"*         AO«     ^O-K^TT. 

4.  4|f.       9.  19¥f.      14.  40ft. 

Page  1O3. 

6.  6?|.     10.  23f|.     15.  ^L. 

13.  f.           is.  3ft.       19.  1426|f. 

1.  9f                         2.   if 

14.  2T3r.        17.  19|.       20.  Iffi. 
15.  21T37.      18.  33|f 

Page  1O7. 

3.  300. 

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4-  is1  =  Iff,  1  =F  ||5,  f  -  Iff. 

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MULTIPLICATION. 
Page  108. 
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Page  1O4. 

2-  f                   13.  42i|. 

4.  141ft.                   5.  81J. 

3.   y  =6.         14.  5ft. 
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SUBTRACTION. 

5.  f  =  If.          16.  5|. 
6.  2i.                  17.  17|. 

Page  1O5. 

7.  51.                  18.  f. 

4-  i-  I  •                      8.  H,  tt- 

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24 


370 


ANSWERS 


Page  1O9. 

DIVISION. 

5. 

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ANSWERS 


371 


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31.  J. 

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89.  The  man 

372 


ANSWEKS 


Page  124. 

19.  .46875.        26.  .006875. 

90.  2|.                  96.  17$. 

20.  .65625.         27.   .01171875. 

91.  1TV¥-               97.  5,  1800. 

21.  .796875.       28.  .135546875. 

92.  39.                  98.  371. 

22.  .78515625.  29.  .0001. 

93.  20.                  99.  2700,  3000. 

23.   .125.             30.  .222464. 

94.  fa.                100.  150. 

24.  .00875.         31.  .05795+. 

95.  42f. 

25.  .2976.          32.  .04707  +. 

DECIMALS. 

4.     1.  .33331           11.  .7435ff. 

2.  .5555f.           12.  1.331. 

Page  131. 

3.  .3846TV         13.  ,37932%. 

2,     1.  &.                 12.  i- 

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9.  ^W              20.  ^M^o- 

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1.  16.66f.                9.  5.3244f. 

11.     2^7- 

2.  35.8.                  10.  48.64. 

3.  .9325.                11.  31.08. 

Page  132. 

4.  4.52.                  12.  .0003158—. 

:2.  1.  i.         8    2To         15-  66I- 

5.  .3475.                13.  3.00625. 

2.  |.           9.  fa.           16.  25f 

6.  76.331.               14.  4627.6428+. 

3.  ^.       10.  TV           17.  50TV 

7.  98.5.                   15.  1899.4839. 

4.  £.         11.  5}f.         18.  100$. 

8.  .54875. 

5.  i.         12.  12T3¥.       19.  700f. 

ADDITION. 

6.  i.         13.  33¥L.       20.  10001. 

Page  135. 

7.  |.         14.  55|.         21.  33f. 

1.  380.246.            11.  142.4430. 

2.  122.995.             12.  126.205. 

Page  133. 

3.  5.125.                13.  1222.18905. 

2.     1.  .25.                  10.  .6. 

4.  6.730.                14.  17.1207. 

2.  .75.                  11.  .375. 

5.  746.58525.         15.  75.225. 

3.  .625.                12.  .8. 

6.  117.766.             16.  529.0625. 

4.  .875.                13.  .0625. 

7.  108.455.             17.  51.170. 

5.  .3125.              14.  .15. 

8.  745.707.             18.  17.2737. 

6.  .4375.              15.  .85. 

9.  787.428.             19.  11.22211. 

7.  .9375.              16.  .52. 

10.  87.474.              20.  1110.00011. 

8.  .53125.            17.  .35. 

1.  555.52.                 2.  1.9375. 

9.  .44140625,,      18.  .2875. 

3.  47070000.960041008. 

ANSWERS 


373 


Page  136. 

21.  8.064.               31.  150. 

4.  5.804.                   5.  .022987875. 

22.  6.963744.         32.  604. 
23.  42.3.                 33.  .0149935. 

SUBTRACTION. 

24.  129.6.               34.  .0000015984. 
25.  52.34375.         35.  6.1625. 

1.  67.98.                   3.  6703.5342. 

26.  1.5625.             36.  3700. 

2.  57.261. 

27.  97.65625.         37.  234.61875. 

Page  137. 

28.  .42624.             38.  3329.6095. 

4.  7.7628.                  6.  5.175. 

29.  4.110092.         39.  .00009. 

5.  4256.84436. 

30.  .65964.            40.  10.41797537. 

3.  36.31119.              10.  991.9001. 

4.  9.8202.                  11.  11.131. 

Page  14O. 

5.  210.8561.              12.  .0000756. 

2.  1.  87.        5.  4069.    9.  3.6. 

6.  295.4526.              13.  17.705. 

2.  .069.     6.  .94.     10.  854300. 

7.  .00009.                 14.  1963.626. 

3.  9560.    7.  92.       11.  10018.2. 

8.  684.999.                15.  .9257926. 

4.  4.53     8.  749.     12.  76541000. 

9.  115.001.                16.  .4234. 

1.  1.  4923.375.         5.  166.11. 

17.  1.  84.655.             4.  106.524. 

2.  1707.                6.  375. 

2.  .495.                 5.  4.99999995. 

3.  138.1875.        7.  945. 

3.  2319.67.           6.  2.0625. 

4.  121.54. 

1.  2512.50.                3.  2519.98. 

2.  706.13. 

Page  141. 

Page  138. 

8.  547.                      2.  142.39. 

4.  .3.                        5.  James,  .3255. 

9.  25313.225.            3.  .000188. 

1.  14.8874.              4.  .637235. 
2.  9.625.                  5.  47.07. 

DIVISION. 

3.  289.7892. 

Page  142. 

1.  1.  3600.                14.  .958. 

MULTIPLICATION. 

2.  .289.                15.  4.6737. 

Page  139. 

3.  784.                 16.  .025. 

2.  1.  32.67.              11.  .020265. 

4.  .01704.             17.  3. 

2.  .3267.              12.  15.1296. 

5.  .51.                  18.  5. 

3.  2.86268.           13.  34.4576. 

6.  7.6.                  19.  30. 

4.  .4077.              14.  .0006076. 

7.  17500.             20.  1000. 

5.  6.0088.            15.  4.08. 

8.  11.2195.           21.  50. 

6.  27121.5.          16.  4.0073328. 

9.  8.76.                22.  3.331. 

7.  27148.6215.     17.  40.12. 

10.  34.6.                23.  .00008. 

8.  .273.                18.  .0001403. 

11.  .01.                  24.  .000005. 

9.  150.                 19.  86213. 

12    9.58.                 25.  .96. 

10.  .20056.            20.  79.88904. 

13.  2. 

374 


ANSWERS 


3. 

1. 

61.544. 

4.  1.08096. 

Page 

147. 

2. 
3. 

32.185. 
45.0167966  +  . 

5.  .005873. 
6.  2500. 

1. 

1.  63900. 
2.  368. 

6. 

7. 

666700. 
34265. 

3.  2402. 

8. 

4751.0875. 

Page 

143. 

4.  690.9. 

9. 

84400. 

2. 

1. 

5.3479. 

6. 

.495674. 

5.  3029. 

10. 

977. 

2. 

.492568. 

7. 

.00000038649. 

2. 

1.  16.16. 

6. 

1197.297. 

3. 

.0249653. 

8.  .000082253. 

2.  2002.036. 

7. 

474.4038. 

4. 

.05908. 

9.  .9391. 

3.  8.15418. 

8. 

5820.048. 

5. 

.00007156.  10. 

0785437. 

4.  1009.77. 

9. 

418.1904. 

5.  2791.425. 

10. 

32904.06. 

Page 

144. 

Page 

148. 

1. 

1. 

.02. 

8. 

.00725. 

5. 

$15.775. 

10. 

$35. 

2. 

.193. 

9. 

1.35. 

6. 

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11. 

$693.545. 

3. 

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10. 

18.5. 

7. 

$7.25. 

12. 

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4. 

4.1. 

11. 

9.35. 

8. 

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13. 

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5. 

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12. 

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9. 

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14. 

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6. 

31.565. 

13. 

9.4625. 

7. 

.0116. 

Page 

149. 

2 

117.50. 

3 

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16. 

5.96. 

19. 

258.408. 

17. 

51.43. 

20. 

20.1575. 

Page 

145. 

18. 

65.487. 

21. 

565.0626. 

i 

1. 

494307. 

6. 

594090620. 

1. 

1000. 

9. 

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2. 

486766. 

7. 

56433692. 

2. 

15. 

10. 

116. 

3. 

59618322. 

8. 

91619232. 

3. 

1.06f. 

11. 

1.002002  -f 

4. 

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9. 

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4. 

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5. 

456462492. 

10. 

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5. 

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13. 

30855.8. 

6. 

27.534. 

14. 

7.706. 

5. 
6. 

$486.08. 
$23040. 

7. 
8. 

75.2735. 
2384.64. 

7. 
8. 

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2.02024. 

15. 

50. 

9.  6494540461.725  +. 

4. 

1. 

594580. 

9. 

5508825. 

Page 

150. 

2. 

459048. 

10. 

4958523. 

16. 

1174.245. 

3. 

976005. 

11. 

18063864. 

17. 

5000000000. 

4. 

3604528. 

12. 

43626114. 

18. 

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5. 

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13. 

119227980. 

19. 

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6. 

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14. 

319868736. 

20. 

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7. 

12439308. 

15. 

452023875. 

21. 

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8. 

4529385. 

16. 

22,891,152. 

22. 

w- 

ANSWERS 


375 


23.  74.76. 

Page  159. 

24.  143869176. 

7.  539951.          14.  8,  6,  3f  . 

25.  ^Mf 
26.  292.25. 

8.  189f.              15.  5,  8,  2f. 
9.  3151.              16.  150000. 

27.  1.12. 

10.  36.                  17.  1728. 

28.  .0125,  .125,  .0025,  fa  1,  ¥^. 

11.  90.                  18.  100000. 

29.  9929.4. 
30.  2.25. 

12.  411f.              19.  190. 

13.  17,  9,  11.       20.  1980. 

31.  494.88. 

32.  $0.50,  $50. 
33.  492.61875. 

Page  16O. 

34.  $170. 

irx» 

35.  If 

5.  2,  1,  3. 

36.  .0793295. 

6.  212.85. 

Page  151. 

7.  19.57TV 

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37.  64278.72.          46.  224fi. 

Q      *7  8  3 

38.  .0225.               47.  $212.75. 
39.  1335.                 48.  102f  |. 
40.  426  075.            49.  $646.95. 

10.  20f|. 
11.  120|. 
12.  62¥7¥97. 

41.  486.                  50.  yW^V 

13.  4584. 

42.  5.125.               51.  $6.22. 
43.  91ff                 52.  $10.831 

14.  62^. 
15.  £26.  9s. 

44.  19.8.                  53.  $28.33i. 
45.  16.50.                54.  $62  50. 

16.  $126.529. 
17.  £116  10s.  2d.  2  far. 

BILLS. 

18.  £81.0541  +. 

Page  153. 

19    £9.9866+. 
20.  $175.194. 

1.  58044. 

21.  £26  11s. 

Page  154. 

22.  £200. 

2.  22  94.                   3.  2167.16. 

Page  161. 

Page  155. 

23.  £250000.           28.  $742.278. 

4.  30.95.                   6.  231.34. 

24.  108440.             29.  25160.62. 

5.  1204.93.               7.  29.74. 

25.  $35.96.              30.  59.68. 

26.  $48.567.            31.  T^. 

COMPOUND   NUMBERS. 

27.  $372.49.           32.  18s.  4d. 

Page  158. 

Page  162. 

4.  1.  486.                  4.  362. 

1.  209. 

2.  38856.              5.  3587. 

2.  8903. 

3.  8480.               6.  89074. 

3.  1452. 

37G 


ANSWERS 


4.  316800. 

22.  1.1  A. 

5.  27732. 

23.   20. 

6.   198. 

24.   14,  50,  12  J,  5,  92;    or,  14,  50, 

7.  63360. 

13,  1,  10. 

8.   15854. 

2e      170  508 
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9.   2  m.  4  fur.  34  rd.  2  yd.  2  ft. 
10.  426,  0,  11. 

1-  3  A.                        7.  3520,  $352. 

11.  47,  141,  3,  1  ft.  6  in. 

2.   128.                        8.  66|. 

12.  1  ra.  1  fur.  31  rd.  12  ft.  3  in. 
13.  40000. 

3.  37£.                        9.  80. 
4.  36.                        10.  $32848. 

14.  77400. 

5.  1728.                     11.  44.20. 

Page  163. 

6.  224^2_                  12.  $85.83|. 

15.   9748. 

Page  167. 

16.   5f. 

13.  $80.65.                19.  4500. 

17.  9f£. 

14.  30.                       20.   Neither. 

18.   ll£f 

15.  $47.11.                21.   12. 

19.   126.631°. 

16.  $1913|.                22.   $480. 

20.   5  ft.  2  in. 

17.  72.                       23.  $425.45. 

21.  3  mi.  57  ch.  63  links. 

18.  $8.50  or  $13.28.  24.  $447.99. 

22.  7603200  or  8763955.2. 

23.   10,  222,  2,  2,  1. 

Page  168. 

24.  501304320;    13925120. 

25.   1125.                    26.  2250. 

25.  75  ft. 

VOLUME. 

Page  165. 

Page  169. 

1.  8650.    '          7.  85,  129,  8,  3,  108. 

1.  730960.                 9.  9872. 

2.  18876.            8.  20275956. 

2.  15,  18,  16.          10.  68^. 

3.  257276^.        9.  7.15|,  8.32. 

3.  55410.                  11.  59772. 

4.  9823032.      10.   198880- 

4.  21870£.                12.  9£. 

5.   15328440.    11.  20. 

5.  60.                       13.  2441. 

6.  13551.          12.  Ifi. 

6.  80.                       14.  36|f. 

7.  36f|i.                  15.  6,  3,  960. 

Page  166. 

8.  56. 
Page  170. 

15.   114,  8,  5,  113f. 

1.   ].   157£  cu.  ft. 

16.  11,  3,  13£. 

2.  20|i  cu.  ft. 

17.  4. 

3.  6f. 

18.   16000000. 

4.  21. 

19.  253,  7,  6,  146. 

5.   1  cu  in. 

20    128. 

6.  176  cu.  yd. 

21.  64040053  sq.  1. 

7.  1777|.  cu.  yd. 

ANSWERS 


377 


Page  171. 

MEASURES  OF  CA- 

8. 101^  cu.  yd.       10.  64  cu.  in. 

PACITY. 

9.  216  cu.  in. 

1.   1.  28.                  5.  154||, 

2.  Vol.  of  a  cd.  ft.  =  1  X  4  X  4. 

2.  64.                  6.  176. 

3.  9/¥. 

3.  296.                7.  5264. 

4.  410J. 

4.  99f.                8.  9.484. 

5.   19683. 

6.  592704. 

Page  175. 

7.  65if- 

9.  ¥V                   10.  255. 

8.  $611.11. 

9.   18. 

2.  1848.                     3.  21.5+- 

10.  40. 

1.  410.36.                  5.  5263ff. 

11.  192. 

2.  28.26.                    6.  842TV 

12.  4f. 

3.  458419^.                7.  1795ff. 

13.  $3588.75. 

4.  7.525.                     8.  45V 

14.  6561. 

15.   1280.  No  allowance  for  corners. 

APOTHECARIES'    MEAS- 

16.  178,687. 

URE. 

Page  172. 

Page  176. 

17.  $1155.               20    4  X  11^ 
18.  6if.                    21.  633600. 
19.  3. 

1.   1536.                  6.  .030273. 
2.  3.                       7.  17,208. 
3.  1584.                 8.  214f 

4.  6f.                      9.  268,740. 

5.  2,112,660.        10.   137,7,2,  5,47. 

Page  173. 

1.  1.  12.                    6.  90. 

DRY   MEASURE. 

2.  12J.                  7.  525. 
3.  11.                    8.  20. 

Page  177. 

4.  11.                   9.  180. 

1.  6.                           6.  831. 

5.  18|.                10.   lOf 

2.  20.                          7.  2443. 

01                                                     Q       01 

2.  53J.                       7.  $180.88. 
3.  1306|.                   8.  74f. 
4.  1TV                       9.  $17.424. 

O.      1.                                                   O.      -^  1  . 

4.  33.                         9.  251. 
5.  28/2-                    10.  319. 

5.  25|.                     10.  13^. 

6.  $17TV                 11.  $20.57. 

CAPACITIES. 

1.   17203.36,  45158.82. 

2.  12.877. 

Page  174. 

3.  746,496. 

12.  $60.48. 

4.  393|. 

378 


ANSWERS 


5.  4.32. 

6.  18.43. 

7.  42.19. 

8.  75.2. 

9.  185.4. 

10.  7,840,800. 

11.  8.671. 

Page  178. 

12.  217.718.  16.   1.24+. 

13.  24.323.  17.  7.5  nearly. 

14.  746f.  18.  131. 

15.  497|.  19.  87+,  70  nearly. 

MEASURES   OF  WEIGHT. 
Page  179. 

1.  6000.  4.  2.97. 

2.  10,406.  5    64,000. 

3.  20,327.  6.  5.203  T. 

"WEIGHTS  AND  VALUES. 

1.  16.50.  5.  1.30. 

2.  2.89.  6.  80. 

3.  64.00.  7.  32,000. 

4.  63,000.  8.  15. 

Page  ISO. 


9.  The  same. 
10.  35.64. 

13.  63521. 
14.  26^- 

11.  47.60. 
12.   150. 

15.  $301. 

1.   1.  228. 
2.  81. 
3.  404,632. 

5.  34,450. 

7-  1™ 
8.  5760. 

Page  181. 

2.  7000.  6.  11T8T°7;V 

3.  1240.  7.  1.28f. 

4.  Lead,  1240.          8.   123.274. 

5.  152TV 

1.  437£.         2.  480.         3.  480. 


1.  32  Ib. 

2.  12  Ib. 

3.  81 63. 

4.  8J  Ib. 


Page  182. 

5.  53179. 

6.  791  Ib. 

7.  98,277  gr. 

8.  18flb. 


L  93  43  °B  7  gr.    5. 

2.  100. 

3.  6-f. 

4.  1400. 


6.  9if. 

7.  308.75. 
8    164.16. 


MEASURES  OF  TIME. 
Page  183. 

I.  86,400.  2.   1  da. 

Page  184. 

3.  10,080.      5.  6176.      7.   117,161. 

4.  2.  6.  4¥V        8.  24T\V 

1.  13.  49.  56. 

2.  13.  46.  38. 

3.  155. 

4.  117. 

5.  Summer. 

6.  1600,  1660,  1776. 

7.  27;  529. 

8.  4  da.  16  hr. 

9.  168. 
10.  i 

II.  Sept.  15. 

12.  24. 

13.  744. 

14.  38  da. 

15.  May  26,  Aug.  31. 

CIRCULAR  MEASURE. 

Page  185. 
1.  1.  3385.  5.   128,939. 

2.  63f.  6.  27-.  7139. 

3.  1416.  7.  527,993. 

4.  14T\.  8.  4-545. 


ANSWERS 


379 


2.  10,800. 

5.     -jfy. 

3.  324,000. 

6-   sV 

4.  f  . 

7.  YTO¥< 

8-  f^-s- 

6.   1  hr.  1  min.  1  sec. 

9.  .000225. 

10.  W*?- 

COUNTING. 

11-  Winr- 

Page  186. 

12.  TiT. 

1.  6480.                    4.  60. 

13.  7V 

2.  18J.                      5.  4800. 

•  "ff*** 

3.  20,736. 

1°'    TSTflTTT' 

1.  100.                       5.  70. 

!"•    TnrnTTF- 

2.  26.40.                   6.  2770. 

17-    T*ffv*. 

3.  Lost  $2.00.          7.  1260. 

19.     TTTrBTJ-' 

4.  60. 
Page  187. 

20.  .00056078125. 

8.   .08£.          9.   12.          10.  93,750. 

FRACTIONAL  RELA- 

Page 188. 

TIONS. 

1.  1.  IfJ-  pt.               8.  9,  11,  .96. 

Page  19O. 

2.  2£  min.             9.  13,  8,  2.56. 
3.  3  yd.  2  ft.       10.  8,  17.28. 

3.  1.  ft.                   9.  Iff. 
2.  •j-ff-j.              10.  ^. 

4.  2.08  gills.        11.  320,  6. 

3.  ^¥9^.              11.  i. 

5.  8,  57,  2f        12.  2,  3,  3.2. 

6.  43,  19,2,  36.  13.  12,  259.2. 

5'  T^               13     e  \5' 

7.  .448.                14.  1,  7,  18. 

Q      ^65                            ^'    fg?' 

2.  1.  If                    9.  2|. 

7.  f.                   15.  X%V 

2.  &.                   10.  23TV 

3.  6,  13,  8.          11.  13,  20. 

4.  5,  13,  5,  6.      12.  4,  2,  2|f. 
5.  1,  5,  If          13.  62,  8. 

DECIMAL  RESULTS. 

6.  88,  26,  8.        14.  2,  4. 

Page  190. 

7.  1.476.              15.  165. 

5.   1.  .21423  -f  .       5.  .49375. 

8.  5qt.                 16.  85ird. 

2.  .01635.             6.  .3375. 

3.  .861  +.           7.  .497  +. 

REDUCTION  ASCENDING. 

4.  .174. 

Page  189. 

Page  191. 

2.  1.  |XiX$Xi  =  *$*. 

8.  .09816  +.      12.  .36875. 

2.  ssVW- 

9.  .875.               13.  .434 

3.  T^ta. 

10.  .122  +.          14.  .5625. 

4.      Yyi^. 

11.  .038—. 

380 


ANSWERS 


REVIEW. 


1.  1.  106.08. 

2.  684.75. 

3.  42. 

4.  52.92. 

5.  4.97. 

2.  21. 

3.  525600. 

4.  15.3125. 
6. 


6.  3. VJ 

7.  535. 

8.  128. 

9.  17.50 
10.  423.36. 

6.  8000. 

7.  31556929. 

8.  62|. 

9.  9892800. 


Page  192. 


10. 
11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 
22 
23. 
24. 

25. 
26. 

27. 


88. 

90°. 

1.4. 


2090. 
6.57. 
17,  14,  33. 

58928f  Ib. 
222fi 

IW? 

836.64. 
.581 
1584. 
$15.50. 


49ff. 

336. 

102f. 


Page  193. 

28.  302.     30.  80. 

29.  941      31.  .6219  -f. 


ADDITION. 
Page  194. 

1.  2,  129,  0,  1,  2. 

2.  27,  213,  2,  1,  0. 

3.  429,  72,  17,  4,  V2. 

4.  96,  3,  9,  4. 
6.  57,  7,  31,  8. 


Page  195. 

6.  16.  6.  2.  1.  10. 

5.  1.  124  rd.,  1  ft.  4.83  in 

2.  18.1808  sq.  in. 

3.  16s.  3.975cL 

4.  7  cwt.  17  Ib.  10.6  oz. 

5.  8  oz.  6  pwt.  2f  gr. 

6.  16#ff  da. 

7.  60Ty^rd. 

8.  40ff. 

9.  13s.  3  iid. 

10.  2  da.  15  hr.  50  min.  35  sec. 

11.  96  rd.  14  yd.  1  sq.  ft. 

12.  27  cwt.  91  Ib.  12  oz. 

13.  2  hhd.  17  gal.  2  qt.  0  pt.  3  gi. 

14.  £5  15s.  8}d. 

6.  .8  crown.  8.   13  ft.  2.73  in. 

7.  .095575. 

Page  196. 

1.  2.  £7  18s.  lid. 

3.  8  A.  1 50  sq.  rd.  11  sq.  yd.  7  sq. 

ft.  104  sq.  in. 

4.  1  T.  14  cwt.  18  Ib.  15  oz. 

5.  183  Ib.  2  oz.  2  pwt.  3  gr. 

6.  17  yr.  41  wk.  5  da.  23  hr.  58 

min.  59  sec. 

7.  Is.  26°  30'  46^". 

8.  41  gal.  1  qt.  1  pt.  3  gi. 

9.  61b.  103  53  1^. 

10.  4  T.  1989  Ib.  8  oz. 

11.  6  pwt.  15  gr. 

Page  197. 

12.  2  oz.  3  pwt.  3  gr. 

13.  11  hr.  59  min.  20  sec. 

14.  16  gal.  2  qt.  0  pt.  3|i  gi. 

15.  21  hr.  24  min. 

16.  4qt. 

17.  3  oz.  5  pwt. 

18.  4  sq.  yd.  6  sq.  ft.  108  sq.  in. 


ANSWERS 


381 


19.  44  sq.  yd.  8  sq.  ft.  99  sq.  in. 

20.  327°  16'  21T9T". 

23.  9.  4.  2. 

24.  18.  2.  11. 

25.  67.  7.  13. 

26.  3  mo.  21  da. 

27.  3.  11.  28. 

Page  198. 

28.  7.  9.  1. 
30.  5.  3.  1. 

2.  206,  5,  11,  8. 

3.  16,  19,  71,  8. 

4.  56,  5,  4,  0. 

5.  77,  6,  9,  0. 

6.  474,  6,  0,  216. 

7.  276,  10,  7,  0. 

8.  200,  19,  86,  8. 

9.  361,  3,  13,  5,  0,  6. 

10.  423,  8,  4,  9. 

11.  544,  4,  7,  2,  16. 

12.  98,  3,  22,  5,  0,  72. 

13.  13,  20,  5,  36. 

14.  317,  0,  1,  2. 

15.  234,  18,  5. 

DIVISION. 
Page  199. 

1.  16,  2,  38. 

2.  2,  0,  71. 

3.  3,  3,  2,  0,  8|. 

4.  12,  17,  25,  12. 

5.  38,  142,  24,  6,  108. 

6.  68,  2,  0,  If 

7.  1,  775. 

8.  10,  26,  0,  2,  1$. 

9.  7,  16,  5. 
10.  5,  3,  11. 
12.  8. 

13. 


14.  890f. 


15.  65f 

16.  4200. 

17.  16. 


Page  2OO. 

18. 
19. 

20. 


LONGITUDE   AND    TIME. 
Page  201. 

3.  45°  21'  27"  ;    3  hr.  1  min.  25| 

sec. 

4.  47°  16'  45". 

Page  2O2. 

5.  87°  23'  45". 

6.  74°  0'  2". 

7.  5°. 

8.  55  min.  56  sec.  past  1  P.M. 

10.  5  hrs.  5  min.  32  sec. 

11.  53  min.  35T75  sec.  past  4  p  M. 

12.  40  min.  58  sec.  past  noon,  July  5. 

STANDARD  TIME. 
Page  2O4. 

2.  15  min.  46  sec. 

3.  Noon. 

4.  19  min.  20  sec.  past  noon. 

5.  1  min.  1  sec.  before  12. 

MISCELLANEOUS. 

1.  66|ct. 

2.  Eastward  ;  21°  15'. 

3.  6  mo.  23  da. 

4.  3510. 

5.  25,  1,  7,  1&. 

6.  42  min.  54^  sec.  past  noon 


7.  1687|. 

8.  Noon. 

9.  324  bu. 
10. 


Page  2O5. 


382 


ANSWERS 


11.  13165f 

Page  211. 

13.  IJft. 
14.  375. 

1.  200.                     8.  $78.24. 
2.  11.25,  78.75.      9.  162.50. 

15.  2.43|. 
16.  50.808. 

3.  1069.20.            10.  $.90. 
4.  48.15.                11.  49.33,  937.27. 

17.  5  Ib.  12  oz. 
18.  iff  f. 

19.  .  00662  bbl. 

5.  180,  360.           12.  6^,  774^. 
6.  144.90.              13.  66. 
7.  2100. 

20.  .82285+. 

Page  212. 

21.  37|. 
22.  4,  55,  37.     London. 
23.  15. 

14.  1500,  200,  300. 
15.  33£$;  33333.     ^]  333. 

24.  5Jfe. 

25.  $16.6272. 

THE   RATE. 

Page  2O6. 

2.  1.  20$.               11.  25$. 

26.  $1,833,500. 

2.  \\%.               12.  28f$. 

27.  36°  8'  40f  f  ". 

3.  37£$.             13.  8£$. 

28.  120. 

4.  lf$.               14.  75$. 

29.  1.50. 

5.  125$.             15.  45$. 

30.  123  A.,   120  sq.  yd.;    B,   148 

6.  31^$.            16.  35$. 

A.,  80  sq.  rds. 

7.  28$.               17.  4}£$. 

8.  97$.               18.  3f$. 

9.  6$.                 19.  |f$. 

PERCENTAGE. 

10.  7$.                 20.  10$. 

Page  21O. 

2.  1.  21.                    16.  80. 

2.  3.                      17.  350. 

Page  213. 

3.  9££.                  18.  8100. 

1.  20$.                      9.  8£$. 

4.  6.                      19.  1400. 

2.  40$.                    10.  700$. 

5.  300.                  20.  800. 

3.  1$.                      11.  7T^$. 

6.  40.                   21.  750. 

4.  5$.                       12.  30f$. 

7.  8.                     22.  1000. 

5.  3|$.                     13.  m\%. 

8.  48.                   23.  1500. 

6.  62£$.                   14.  17£$. 

9.  fci.                   24.  3000. 

7.  150$.                  15.  4£$. 

10.  35.                   25.  f 

8.  4£$.                    16.  25/7$. 

11.  40.                   2o.  TV 

12.  320.                 27.  100,000. 
13.  60.                   28.  144. 

Page  214. 

14.  300.                 29.  9. 

17.  20$.              19.  50$. 

15.    .75.                 30.  $21.42. 

18.  33|$.           20.  $675,  19|^$. 

ANSWERS 


383 


THE  BASE. 

1.  13240.  74^27.            5.  280;  350. 

2.  1.  3080.                  7.  2100. 

2.  6140.                       6.  $1.20. 

2.  3600.                  8.  6600. 

3.  800.                         7.  168. 

3.  1666f.                9.   72. 

4.  48f                         8.  5.00. 

4.  24.75.               10.  105. 

5.  $643.60.           11.  900. 

Page  218. 

6.  296£f.              12.  160. 

9.  64000.                 13.  50000. 

10.  6f  loss.               14.  400. 

Page  215. 

11.  3952.                   15.  3  Ib. 

13.  162.                20.  12£. 

12.  551A.55sq.rd.  16.  15  cts. 

14.  128.               21.  400. 

15.  96.                 22.  $1.60. 

Page  219. 

16.  140.                23.  150. 

1.  $1.72|;  $12.07|;  8.62£. 

17.  288.                24.  1200. 

2    375. 

18.  160.               25.  13950. 

3.  1968;  2214. 

19.  12£.                26.  15,010,000. 

4.  9060;  1510. 

3.  750.          4.  256.          5.  518.40. 

5.  8%. 
6.  145.80;  16345.80. 

1.  100;  115.               6.  1700. 

7.  177|$. 

2.  400.                         7.  10000. 

8.  9JT$  ;  1584. 

3.  2680.                      8.  7692. 

9.  21if  %  ;  1240. 

4.  1300000.                 9.  $35. 

10.  171.5. 

5.  551  A.  55  sq.  rd.  10.  6300. 

11.  3456;  3784^;  3127^. 

12.  13400;  3082. 

Page  216. 

13.  $41.771;  $12.531. 

11.  10000. 

14.  \\%  ;  10120. 

Page  217. 

15.  800;  72. 

3.  1.  4200.                  9.  505^. 

17.  f  ;  2f  . 

2.  400.                 10.  1^. 

18.  45$. 

3.  738.                 11.  1200. 

19.  25$. 

4.  564.                  12.  848. 

20.  20$  ;  120. 

5.  1120.                13.   225. 

21.  31*|  %  ;  945. 

6.  14000.              14.  2|£f. 

6  3  /u  ) 

7.  600.                  15.  262J. 

1.  16f$. 

8.  3807^.           16.  1882f$f 

4.  1.  1600.                  6.  $700. 

Page  22O. 

2.  2755f.                7.  112. 

2.  $6247.50.               6.  95/T. 

3.  1890.                  8.  525. 

3.  $200.                      7.  68f$. 

4.  5100.                  9.  |f. 

4.  19^$.                  8.  261f 

6.  2750.               10.  720. 

5.  88.                         9.  41TW$. 

384 


ANSWERS 


10.  |16000.                 13.  40%  ;60%. 

Page  225. 

11.  800.                       14.  121f|%. 
12.  9900.                     15.  lli%. 

12.  75;  250. 
13.   16|  %  ;   2  cts. 

14.  $0.01T\;$0.106£. 

COMMERCIAL    DIS- 

15. 25;  331%. 

COUNT. 

16.  25;  25%. 

17.  $87.50;  162.50. 

Page  222. 

18.  .50;  28f%. 

3.  $10.91£;  $79.08f. 

19.  1;  16|%. 

4.  $4.40. 

20.  1.50;  1.62. 

5.  $478.40. 

21.  100;  110. 

6.  50  off  =  $1000  off;  25  and  25 

22.  150;  30. 

off  =  $875  off. 

23.  212.96/7;  17.03£f 

7.  $573.75. 

24.  $.935;  $5.50. 

8.  $1778.65. 

25.  7|%;  $54. 

9.  $11.63. 

26.  630;  677.25. 

10.  $2694.76  +. 

27.  $7692;  7653.54. 

11.  $66.69. 

28.  $2.36;  77.88. 

12.  $390.91. 

29.  71%;  $27. 

13.  35.4%. 

30.  10.775;  1928.725. 

14.  $30.78. 
15.  The  former. 

1.  $3450.                 4.  $4.12^-. 

16.  $6.00. 

2.  25%.                   5.  100%. 

3.  $300.                   6.  40%. 

1.  37%.            3.  54|%. 

2.  49%.            4.  27£{f#. 

Page  223. 

Page  226. 

18.  $55. 

7.  $24.                     15.  121  cts. 

19.  $389.54. 

8.  $1.66J.                16.  26i%. 

20.  $4.80. 

9.  12f%.                 17.  $50. 

10.  $1.20.                 18.  $250. 

11.  33i%.                 19.  25%. 

GAIN  AND  LOSS. 

12.  $1150.                 20.  12|f. 

Page  224, 

13.  57  cts.                 21.  66|. 

14.  $65. 

6.  2100;  12600. 

7.  48;  5.52. 

8.  2140;  25%. 

9.  45;   51.30. 

Page  227. 

10.  105;  805. 

22.  $5.50.                 24.  10f%. 

11.  1;  20. 

23r  75  cts.                25.  $31.25. 

ANSWERS 


385 


COMMISSION. 

Page  232. 

Page  229. 

17.  28^%.               23.  10|%. 

18.  $aV                     24.  The  latter. 

6.  $45.50. 

19.  25%.                   25.  $1.25. 

6.  2}%. 

20.  20%.                   26.  $1194.125. 

7.  $170.73;  $6829.27. 

21.  $21.60.                27.  $30000. 

8.  $6437.50. 

22.  6|%.                  28.  $58.87. 

9.  $3932.04. 

10.  $8.91. 

11.  $4120. 

Page  233. 

12.  $753. 

29.  $2266.66f  ;  2833.331;  2436.66| 

13.  $530. 

30.  $2000. 

14.  $64. 

31.  3%. 

15.  $6.88;  $165.12. 
16.  $4662. 

STOCKS. 

17.  $2776.193. 

Page  235. 

6.  $5137.50.            7.  4f 

Page  23O. 

Page  236. 

18.  3042.65  bu. 

8.  71f. 

19.  $34.83. 

1.  $16612.50;  $600. 

20.  $1920. 

2.  $4700. 

21.  $7443.75. 

3.  $450. 

22.  $218.125. 

4.  3^-f-;    3T7T%  ;    4|ff  %  ;   4^% 

23.  $6489.175;  $252.825. 

5jf  %• 

24.  $9950.25;  $49.75. 

5.  No.  3,  $2424  ;  No.  4,  $1636. 

25.  $600,000. 

6.  No.  3,  5fff  %  ;  No.  4,  3Tyr%. 

26.  $20600  ;  137333^  Ib. 

7.  403  shares  +. 

1.  $162.50-              2.  $450. 

8.  85f 

9.  $1595. 

10.  22  shares. 

Page  231. 

11.  5%  at  60  by  $20.00. 
12.  26812.50. 

3.  Gain,  18|%.        10.  $250. 

4.  10%.                     11.  120000. 
5.  $195.00.                12.  14T6r. 

Page  237. 

6.  56£%.                   13.  25%. 

13.  6%  at  90. 

7.  35T57%.                 14.  $2275. 

14.  225%. 

8.  13%.                     15.  $3891.625. 

15.  13015f. 

9.  Gain,  37+%.     16.  $180.375. 

16.  84. 

25 


386 


ANSWERS 


17.  $594. 

Page  242. 

18.  3732. 

12.  2116.80.              18    \%. 

19.  264. 

13.  13,600.                19.  $42,180. 

20.  3065.20. 
21.  262.50. 

14.  24,500.                20.  $119.10. 
15.  $13,500.              21.  $3676.21 

22.  88|. 

16.  $76,800.              22.  212. 

23.  5118.75. 

17.  $48.                     23.  451.50. 

24.  The  latter. 

25    The  latter,  $50.50. 

Page  243. 

26.  21,050. 

24.  $3168.                 26.  $3516. 

Page  238. 

25.  $3699.50. 

27.  10,000.            34.  $60;  4^. 

28.  5fff.               35.  4134. 

TAXES. 

29.  $13,000.          36.  6780. 

Page  244. 

30.  $6780.             37.  Increase,  $20. 

1.  6  mills.               6.  $27.70. 

31.  121f               38.  58f 

2.  $29.31.                7.  .OOlf. 

32.  352.                39.  15^y#. 

3.  Bfc-                    8.  .003. 

33.   b%. 

4.  $6000.                 9.  3826.53. 

Page  239. 

5.  $10.                   10.  .007. 

40.  71,250.            47.  5000;  22|%. 

41.  5%.                 48.  $4500. 

Page  245. 

42.  621                  49.  224  shares. 

11.  17,500.               13.  $19,100. 

43.  .07^.              50.  109if. 

12.  6000.                  14.  $12,445.13 

44.  14f                 51.   101£. 

45.  300.                 52.  §\%. 

TAX  FROM   TABLE. 

46.  104.                 53.  7s. 

Page  246. 

16.  1.  $13.65.             6.  304.50. 

Page  24O. 

2.  35.34.               7.  480.75. 

54.  200  shares. 

3.  54.                   8.  1673.52. 

4.  147.83.             9.  3015. 

5.  165.96.           10.  8508.08. 

INSURANCE. 

Page  241. 

INDIRECT  TAXES. 

1.  $2400.                   7.  $297,000. 

Page  247. 

2.  $99  40.                  8.  375. 

3.  $750.                   8.  $89.061. 

3.  $57.60.                  9.  The  2d 

4.  $603.75.              9.  $10.62£. 

4.  $337.50.              10.  1%. 

5.  $130  625.           10.  $294. 

5.  $187.50.              11.  $58.39. 

6.  $2480.               11.  $1547, 

6.  1|^. 

7.  $362.  50. 

ANSWERS 

387 

Page 

248. 

Page 

253. 

12 

$158.40. 

3. 

$33.81. 

5. 

$408.82. 

13 

$94.03. 

4. 

$1651.67. 

14 

$1078. 

15 

$591.09. 

2.  1. 

$29.17. 

5. 

$2.00. 

16 

$6629.12. 

2. 

$11.31. 

6. 

$87.03. 

17 

$76500. 

3. 

$2.14. 

7. 

$76.47. 

18 

20$. 

4. 

$3.76. 

8. 

$80.56. 

19 

$.39  +  ;  $391.66. 

20 

$825. 

21 

Page 

254. 

22 

$9000. 

1.  1. 

$100.80. 

6. 

$45. 

23 

$1.40. 

2. 

$15.90. 

7. 

$142.27. 

Page 

249. 

3. 

$7.00. 

8. 

$1077.50. 

24 

$125.              25.  $800;  $.50. 

4. 

$26.67. 

9. 

$2358.73. 

INTEREST. 

5. 

$10.29. 

10. 

$197.05. 

Page 

251. 

2.  1. 

$61.70. 

5. 

$180.80. 

1. 

1.  $8. 

6.  $360. 

2. 

$464.62. 

6. 

$8406.98. 

2.  $24. 

7.  $343. 

3. 

$1317.23. 

7. 

$6008.92. 

3.  $90. 

8.  $384. 

4. 

$843.40. 

4.  $112. 

9.  $486. 

2. 

5.  $150. 
1.  $147.99. 

10.  $800. 
5.  $444.05. 

Page 

255. 

2.  $295.00. 

6.  $172.331. 

4.  1. 

$150.10. 

5. 

160.74. 

3.  $290.50. 

7.  $1297.45. 

2. 

$1486.51. 

6. 

58.912. 

4.  $327.75. 

3. 

$6.15. 

7. 

$318.07. 

Page 

252. 

4. 

$879.75. 

8.  $1942.38. 

10.  $775.83|. 

9.  $621.00. 

Page 

256. 

3. 

1.  $.546. 

6.  $786. 

1.  1. 

$51.13. 

2.  $15.85. 

7.  $270.74. 

2. 

$396.73. 

3.  $4.94. 

8.  $559.65. 

3. 

$1196.81. 

4.  $11.08. 

9.  $427.88. 

4. 

$122.45. 

5.  $18.42. 

10.  $402.17. 

5. 

$196.45. 

4. 

1.  $319.08. 

6.  $358.52. 

2.  $806.94. 

7.  $1198.16f. 

2.  1. 

$764.07. 

3.  $141.37. 

8.  $230.13. 

2. 

$966.93. 

4.  $310.07. 

9.  $127.28. 

3. 

$1313.93. 

5.  $236.88. 

.10.  $208.30. 

4. 

$251.73. 

1. 

1.  $1,20. 

2.  $157.58. 

5. 

$861.15. 

388 


ANSWERS 


Page  258. 

Page  261. 

1.   1.  $750.              11.  $40050.59. 

5.  $110.95.                 7.  $55.94. 

2.  $336.              12.  $2980.87. 

6.  $81.25.                  8.  $607.55. 

3.  $242.42.         13.  $551.11. 

4.  $7159.13.       14.  $2877.63. 
5.  $102.86.         15.  $9664.54. 

ANNUAL  INTEREST. 

6.  $1388.89.       16,  $293.43. 

Page  262. 

7.  $6903.51.       17.  $665.66. 

1.  1.  $106.52.            4.  $277.33. 

8.  $2099.72.       18.  $40396. 

2.  $72.77.              5.  $202.56. 

9.  $7000.00.       19.  $4027.69. 

3.  $1045.80.          6.  $360.58. 

10.  $6222.22.       20.  $773.56. 

2.  1.  $447.36.            4.  $1080.44. 

2.  1.  6%.                  3.  5%. 

2.  $322.778.          5.  $2958.37. 

2.  5%.                  4.   5£. 

3.  $730.056.          6.  $1246.68. 

Page  259. 

Page  266. 

6.  1%. 

19.  1.  April  17,  $637.99. 

6-  10iii%- 

2.  July  22,  $806.517. 

7-    3TT88T%' 

3.  Feb.  6,  $404.49. 

8.  5f%. 

4.  March  21,  $1076.62. 

9.  331%  ;  20/0  ;  16f%. 

5.  Dec.  9,  $635.41. 

10.  28f.%  ;  25%  ;   10%. 

6.  July  17,  $975.228. 

3.   1.  2  yr. 

7.  May  30,  $2654.92. 

2.  |  .yr. 

8.  July  8,  $2856.77.  ' 

3.  2  yr. 

9.  Aug.  11,  1901,  $1116.90. 

4.  4  yr.  8  mo.  14  da. 

10.  Dec.  3,  $1995.16. 

5.  5  yr.  4  mo.  13  da. 

6.   16f  yr. 

Page  268. 

7.   121  yr. 
8.   20  yr.  ;   16f  yr.  ;   14f-  yr. 

1.  $335.99.                 5.  $261.68. 
2.  $122.20.                 6.  $85.23. 

9.  22|  yr. 
10.  331  yr.  ;  28f  yr.  ;   25  yr. 

3.  $71.95.                  7.  $19.08. 
4.  $441.40. 

1.  $10  20.                   4.  2  yr.  10  mo. 

2.  $33,3331.              5.  41^. 

Page  269. 

3.  $1050.92. 

8.  $1330.88.             10.  $5747.13. 

COMPOUND   INTEREST. 

9.  $8039.47.             11.  $914.80. 

Page  26O. 

Page  271. 

1.  $53.58.                   3.  $60.366. 

1.  $1316.76.               3.  257.21. 

2.  $516.05.                4.  $229.17. 

2.  $287.72.                 4.  $267.14. 

ANSWERS 


389 


Page  272. 

Page  277. 

5.  $500.49.                8.  1860.32. 

9.  $1017.60.            10.  $1250.00. 

6.  $5204.67.              9.  $8972.43. 
7.  $2958.49. 

1.  $404.04.               5.  $1966.84. 

2.  $786.55.               6.  $1441.62  +. 

Page  273. 
10.  $1295.71. 

3.  $1287.11  +.         7.  $1025.64  +. 
4.  $7653.06  +.         8.  $154  50. 

BANK  DISCOUNT. 

TRUE    DISCOUNT. 

Page  274. 

Page  278. 

1.  $1.46;  $348.54. 

1.  $377.36;  $22.64. 

2.  $6;  $394.00. 

2.  $183.48;  $1652. 

3.  $6.30;  $533.70. 

3.  §168.09;  $11.91. 

4.  $8.00;  $592.00. 
5.  $50;  $1950.00. 
6.  $1.11;  $79.49. 
7.  $16.25;  $4983.75. 
8.  $6.99;  $773.01. 

4.  $508.47;  $91.53. 
5.  $297.43;  $52.57. 
6.  $1483.31;  $16.69. 
7.  $1760.56;  239.44. 
8.  $403.099;  84.65. 

9.  $10.63;  $589.37. 

9.  $371.806;  50.194. 

10.  $25.83;  $974.17. 

10.  $416.848;  62.527. 

Page  275. 

1.  Apr.  21,  1898;   37  da.;    $3.60; 

Page  279. 

$496.40. 
2.  May   2,    1898;    31   da.;    $4.13; 

1.  $166.09;  33.91.     6.  $40.69. 
2.  $45.                         7.  $1596.815. 

$795.87. 
3.  Apr.    5,   1898;    85  da.;    $5.67; 

3.  $1.355.                    8.  $1408.82. 
4.  $21.56.                    9.  $750. 

$394.33. 
4.  Oct.  20,  1898;  119  da.;  $10.78; 

5.  $1.25.                     10.  $467.29. 

$454.97. 

1.  25|fc.                      2.  66f#. 

Page  276. 

5.  June  6,  1898  ;   24  da.  ;    $14.58  ; 
$3630.42. 

Page  28O. 
3.  $327.24;  $654.48. 
4   94% 

6.  Nov.  24,  1898;  54  da.  ;  $21.897; 
$2411.10. 

«     v^/O* 

5.  14f  yr. 
6.  $2004.51. 

1.  $808.08.                5.  §243.14. 

7.  Dec.  12;  2  mo.  16  da.;  $22.93; 

2.  $999.495.              6.  $987.09. 

$1952.07. 

3.  $3045.69.              7.  $2666.66f. 

8.  $520.25. 

4.  $15,186.03.           8.  $418.00. 

9.  $497.92;  $320.86. 

390 


ANSWERS 


10.  $6262.136. 

Page  29O. 

11.  $2465.625. 

6.  .40.                  14.  162T\52. 

12.  $13.66. 

7.  19.6.                15.   900. 

13.  $1418.33. 

8.  24.                    16.   2.45. 

14.  $813.69. 

9.  1080.               17.  6|. 

15.  $678.37. 

10.  1175.23.          18.  .80  +. 

11.  32.                    19.  33  hr.  20  min. 

Page  281. 

12.  4f.                   20.  40|. 

16.  $32.17.                 20.  $1886.50. 

13.  67TV 

17.  $5300.                  21.  28#. 

Page  291. 

18.  30^oV  bbls.          22.  $92.38. 
19.  $75.76. 

21.  2if.           28.  £227  12s.  Id. 

22.  50.             29.  $57.33. 

EXCHANGE. 

23.  13.             30.  7. 
24.  1095.         31.  2560. 

Page  284. 

25.  42.             32.  101. 

4.  $594.50.                 8.  $6480. 

26.  6|.             33.  2  yr.  9  mo.  10  da. 

5.  $1818.                   9.  $1395.33. 

27.  26HI- 

6.  $2992.50.             10.  $1230.79. 

7.  $3037.50.             11.  $5244.37. 

Page  292. 

Page  285. 

12.  $882.                 18.  $2041.65. 
13.  $8782.81.          19.  $3495.62. 

34.  $2.46+.             38.  900. 
35.  12.                        39.  216. 
36.  10.                       40.  2511. 
37.  74.  66^. 

16.  $6818.95.          20.  $325.05. 

17.  $4600.              21.  $2819.44. 

Page  286. 

COMPOUND  PROPOR- 

22. $1200.00.          23.  $745.15. 

TION. 

Page  293. 

RULE  OF  THREE. 

1.  31.                        5.  473. 

Page  288. 

2.  4.                           6.  $3.84. 

1.  24.            7.  26.4.            12.   13f 

3.  1188.                     7.  6. 

2.  54.            8.  12.               13.  £f 

4.  $131.20.                8.  840. 

3.  32.            9.  3.6.              14.  2.75. 

4.  2.            10.  8.                15.  1.6|f 

Page  294. 

5.  4.            11.  f.                 16.  .0975. 

9.  30.         14.  $800. 

6.  54. 
Page  289. 

10.  32.         15.  1828f 
11.  $600.      16.  81  ten-ounce  loaves. 

1.  42.              3.  53|.              5.  12TV 

12.  125.       17.  "21 

2.  4.                4.  6. 

13.  63.         18.  1430. 

ANSWERS 


391 


Page  295. 

19.  $124.75.  22.   74^  da. 

20.  6027.  23.   9'f  yr. 

21.  lOihr. 

CAUSE  AND  EFFECT. 
Page  296. 


1.  10. 

2.  $10118.77. 


3.  21.  5.  6. 

4.  $100.         6.  10. 


PROPORTIONAL   PARTS. 

Page  297. 

1.  25,  35. 

2.  264,  288,  312,  336. 

3.  360,  240,  180. 

4.  60,  45,  15. 

6.  3149.17^,  4723.76rV,  5511. 

6.  .50,  1.00,  2.50,  5.00. 

7.  3000,  6000,  9000.        « 

Page  298. 

8.  15,  25,  20. 

9.  100,  20,  13£. 

PARTNERSHIP. 

Page  299. 

1.  $78,  $104. 

2.  $84;  $72. 

3.  $67.50,  $72,  $81. 

4.  $179.048,  $537.142,  $223.810. 

5.  $2175,  $2030,  $2755,  $2320. 

6.  $750,  $500,  $250. 

7.  $60,  $72. 

Page  3OO. 

8.  $14,  $28. 

9.  $266|,  $333i,  $400. 

10.  $862.50,  $575.00,  $862.50. 

11.  $1080,  $1600,  $1820. 


12.  f. 

13.  $1.50,  $3.00. 

14.  $48,  $70. 

15.  $6039.04if!f;  $2228.00}-faf. 

16.  $481.25;  $1196.25. 

17.  $710,  $352.80. 

18.  $1000,  $1500,  $2000. 

Page  3O1. 

19.  $560,  $2240,  $2800. 

20.  $273  365  +,  $476.635  +, 

21.  45  lb.,  41  lb.,  i  Ib. 

22.  $332.50,  $525. 

23.  $6000,  $14000. 

24.  $165,  $210,  $225. 

25.  700,  1866f,  933£. 

26.  $101.41,  $98.59. 

27.  $1333.331,  2000,  2666.66f. 

Page  3O2. 

28.  $1000,  $1500. 

AVERAGES. 

1.  426. 19|.  3.  20. 

2.  127. 

Page  303. 
4.   1£  oz.  5.  4  mo. 

AVERAGE  OF  PAY- 
MENTS. 
1.  6TV  2.  7^V 

Page  3O4. 

3.  5  mo. 

4.  2|mo. 

5.  8f. 

6.  May  26. 

7.  10  mo. 


392 


ANSWERS 


8.  1  mo.  12  da. 

9.  4$  mo. 

10.  6/^5-  mo. 

11.  $545.45;   1  yr.  8  mo. 

12.  Oct.  23d. 


Page  3O5. 

13.  Aug.  20.  17.  Aug.  19. 

14.  June  21.  18.  Feb.  3. 

15.  Aug.  22  19.   Apr.  22. 

16.  Aug.  21. 


Page  3O6. 

20.  July  11 ;   Dec.  26.     24.  10  mo. 

21.  3^  mo.  25.  9  mo. 

22.  9  mo.  26.  50|. 

23.  18  mo.  27.  50  da. 


Page  3O7. 

28.  April  15.  30.  i 

29.  31  mo. 


INVOLUTION. 
Page  3O8. 

1.  1,  2,  3,  4,  5,  6,  7,  8,  9,  0. 

2.  P  =  1,  32  =  9,  52  =  25,  72  =  49, 

92  =  81,  102=:100,  152  =  225, 
252  =  625. 

3.  I3  ==  1,  23  =  8,  32  =  27,  43  =  64, 

53  =  125,  63  =  216,  73  =  343, 
83  =  512,  93  =  729,  O3  =  0. 

4.  aj  =  l,aj  =  4,a?  =  9,a?  =  16Ja:  = 

25,  x  ==  36,  x  =  27,  x  —  64, 
x  =  256,  x  =  25,  x  =  125,  x  = 
216. 


5.  x  =  400,  x  =  900,  x  =  1600, 

x  =  2500,  x  —  3600,  x  =  4900, 
x  =  6400,  x  ==  8100,  x  = 
10,000. 

6.  x  =  1000,  x  =  8000,  x  =  27,000, 

«  =  64,000,  x  =  125,000,  x  = 
216,000,   x  =  343,000,   x  = 
612,000. 
7-  (£)2  =  i  (i)2  -  A.  (i)*  =  A. 

a)2  -  tv.  (i)2  -  A.  (t)2  = 


Page  3O9. 

8    (i)3  =  i,  (I)3  =  *r,  (I)3  =  II, 

(I)3  -  Ml- 
9.  .01,  .04,  .09,  .16,  .25,  .36,  .49, 

.64, .81. 

10.  .001,  .008,  .027,  etc. 
11-   (I)2  =  TV   (-4)2  =  -16,  (f)3  - 
Tb.  (*)*=!**>  (-I)4  =  -0001, 
(.02)3  ==  .000008. 

1.  *  =  625. 

2.  a:  =  1225. 

3.  x  •=  7744. 

4.  x  =  10,201. 

5.  a?  =  169. 

6.  x  =  729. 

7.  a?  =  4913. 

8.  a;  =  1000. 

9.  x  =  9261. 

10.  x  --=  ft. 

11.  (.001  )3=:. 000000001. 

12.  153  =  . 003375. 

13.  .043  =  .000064. 

14.  .001953125. 

15.  .000729. 

16.  .00000625 

17.  .005  =  .000000125. 

18.  4.2025. 

19.  (25t)" 


ANSWERS 


393 


20. 

(4.6<Xty)*  ==  20.261286|-  1-  . 

3.  1.  .9354+.       11.  .6123+. 

21. 

468.884869+. 

2.  .9428+.       12.  .9684+. 

3.  .2886+.       13.  1.5. 

Page  312. 

4.  |f.                14.  .5059+. 

1. 

12.              7.  36.              13.  63. 

5.  \l.                15.  1.3462+. 

2. 

14.              8.  35.             14.  72. 

6.  .8291+.       16.  .8164+. 

3. 

16.              9.  42.              15.  80. 

V.  .6928  +.      17.  .488  nearly. 

4. 

18.            10.  44.              16.  91. 

8.  .70716  +.    18.  8.0702  nearly 

6. 

24.            11.  51. 

9.  .86602  +.    19.  10.1. 

B. 

26.            12.  54. 

10.   .7905  +.      20.  .66  +. 

Page  315. 

i. 

2. 

10.            11.  53.         21.  821. 
100.          12.  63.         22.  886. 

SQUARES. 

3. 

25.            13.  67.        23.  972. 

Page  317. 

4. 

3i.            14.  84.        24.  999. 

1.  35.                        8.  $5120. 

5. 

52.            15.  96.        25.  2424. 

2.  45.                         9.  $187.20. 

6. 

83.            16.  127.      26.  2504. 

3.  80  sq.  rd.            10.   100. 

7. 

125.          17.  266.      27.  3546. 

4.  3733.523ft.        11.  62  in. 

8. 

376.          18.  344.      28.  5555. 

5.  23  rd.                  12.  The     rectan- 

9. 

401.          19.  612.      29.  6325. 

6.  12.649  +  rd.                 gular  field. 

10. 

10,000.    20.  607.      30.  6453. 

7.  85.                                 $18.34|. 

SQUARE   ROOT. 

Page  316. 

Page  318. 

1. 

1.  i.                      9.  ff. 

2.  h.  =  3.605  +. 

2.  f-                    10.  f|. 

p.  =3.316  +. 

3-  TV                   H.  Iff- 

b.  =  4.358  +. 

4-  TVV                12-  Iff- 

p.  =  4.8  nearly. 

5.  TV                  13.  |ff. 

h.  =  19.104. 

Gil                              1  A.        1000 
•    2T-                         J*'    TRUTHS' 

3.  40.                          5.  122.4  +. 

7.  TV             is-  HH- 

4.  28.284  +.             6.  30  ft. 

8-  if-                  16.  ffff. 

2. 

1.  .3.                          9.  .306. 

2.  .9486  +.             10.   .315. 

Page  319. 

3.  .12.                      11.  .063. 

4.  .3794.                  12.   1.296. 

7.  44.9  very  nearly. 

5.  .1.                        13.  14.31. 

8.  56.69  +. 

6.  .3513  +.             14.  .0099. 

10.  b  ==  6,  p  =  8. 

7.  .874.                    15.  .0101. 

2.  9.921  +.               4.  99215.67  +. 

8.  .6555. 

3.  242.63  +.             5.  21.33  +. 

394 


ANSWERS 


CUBE   BOOT. 
Page  32O. 

=  3,  for  3  X  3  X  3  =  27. 
f  125"=  5,  for  5  X  5  X  5  =  125. 
^343"=  7,  for  7  X  7  X  7  =  343. 
1^512~=  8,  for  8  X  8  X  8  =  512. 
V 729"=  9,  for  9  X  9  X  9  =  729. 
f  409&  —  16,  for  2s  x  23  X  23  = 

4096. 

if  42875~=  35,  for  53  X  7s  =  42875. 
1^166375   =  55,   for  53    X    11s  = 

166375. 

=  57,    for    33  X   193  = 

185193. 


Page 

324. 

1. 

1.  85. 

12.  32. 

2.  42. 

13.  43 

3.  25. 

14.  39. 

4.  32. 

15.  84. 

6.  47. 

16.  123. 

6.  125. 

17.  325. 

7.  177. 

18.  526. 

8.  126. 

19.  642. 

9.  536. 

20.  1234. 

10.  345. 

21.  2345. 

11.  23. 
Page 

325. 

1. 

2. 
3. 

I-       4.  f. 
i-       5-  I- 
f       6.  H- 

7-   H-        10.  ft. 
8.  |f.        11.  f-f. 
9-  If-        12-  |f. 

1. 

2. 
3. 
4. 

.2. 
.43+. 
.928+. 
.5. 

9.  .464+. 
10.  1.816+. 
11.   1.91+. 
12.  3.27. 

5. 
6. 

.627  +. 
1.3. 

13.  3.45. 
14.  .01. 

7. 

2.1. 

15.  .00464+. 

8. 

5.7. 

1.  .908+.  6.  .87+.  8.  .94-1-. 

2.  .88+.  6.  .94+.  9.  .93+. 

3.  .61  +.  7.  .96  +.  10.  .501  +. 

4.  1.07 +. 

VOLUME. 

2.  72. 

3.  6084  sq.  in. 

4.  3|  sq.  ft. 

Page  326. 

5.  12.9  + in. 

6.  Length,    95.11    in. ;    width  and 

depth,  47.55  in. 

7.  $13.48. 

8.  28.4  + ft. 

SIMILAR   FIGURES. 
Page  327. 

2.  R.  =  V~QO. 

3.  2  :  1. 

4.  D.  =  4. 

5.  L.  =  65.6;  W.  =  41. 

6.  R.  =  17.58. 

7.  l^hr. 

8.  16  bbl. 

9.  6-1- gal. 
10.  $4800. 

Page  328. 

2.  3.  6.  1  :  64. 

3.  27  times.  8.  2.16ft. 

4.  $13500.  9.  23.49  ft. 

5.  1404928. 

Page  332. 

1.  5625  sq.  yd.          4.  3|  sq.  ft. 

2.  600  sq.  rd.  5.  370.75  sq.  ft 

3.  43.30  sq.  ch. 

Page  333. 

1.  25|  A.  3.  f  sq.  ft. 

2.  8|  sq.  ft.  4.  8  A. 

1.  2528  sq.  rd.  2.  10  rd. 


ANSWERS 


395 


Page  334. 

.rd.    2.  20fff  sq.  rd. 


Page  335. 

1.  288  sq.  in.  2.  322.575  sq.ft. 

Page  336. 

1.  65.97  ft.  4.  25.78  sq.  in. 

2.  10.5  -f  yd.  5.   14£  ft. 

3.  100.5312yd. 

Page  337. 

6.  159.315  sq.  rd. 

7.  43.9824  ft. 

8.  .3183  ft. 

9.  153.9384  sq.  yd. 

10.  1105.84  +  sq.  ft. 

11.  12  sq.  ft. 

12.  5500  sq.  ft. 

13.  7.854  A. 

14.  22,500  sq.  yd. 

15.  13,950  sq.  yd. 

16.  144  sq.  ft. 

17.  145.309  sq.  ch.,  or  14.5309  A. 

18.  100.58ft. 

19.  141.372  rd. 

Page  338. 

20.  203.716  A. 

21.  312  A. 

22.  491.189  rd. 

23.  3883.0176  sq.  ft. 

24.  50.2656  sq.  ft. 

25.  4.8744  sq.  ft. 

26.  5.2  very  nearly. 

Page  339. 

1.  1400  sq.  in.  4.  98.9604  sq.  ft. 

2.  1178.1  sq.  in.  5.  46.50  sq.  ft. 

3.  288  sq.  ft. 


Page  34O. 

6.  218.77  sq.  in. 

7.  4417.875  cu.  in.;  144  cu.  ft. 

63,6174  cu.  ft.  ;  15.75  cu.  ft. ; 
105.84  cu.  in. 

8.  144  in.  =  12  ft. 

9.  4.71TVft. 

10.  139.0608  cu.  in. 


Page  341. 

1.  300  sq.  ft,          4.  .24  gal. 

2.  17.6715  sq.  ft.    5.  659.2512  cu.  in. 

3.  42.07  bu. 

Page  342. 

1.  112£sq.  ft.  4.  81.8cu.  ft. 

2.  1510.95  sq.  in.       5.  $167.80. 

3.  4052.664  cu.  in. 

1.  1017.8784  sq.  in. ;  3053.6352  cu. 

in. 

2.  452,3904  sq.  in. 


Page  343. 

3.  65.45. 

4.  1767.15  cu.  in. ;   706.86  sq.  in. 

5.  339.2928  cu.  in. 


1.  1/2. 

2.  3.1416  X  V 


3.  5  X  .7071. 

4.  12.50;  19.64. 


Page  344. 

1.  1092  cu.  ft. 

2.  636.174  cu.  ft. 

3.  314.16  sq.  ft.  ;  392.70  sq.  ft. 

4.  1809.5616  sq.  in. 

5.  18.13  gal. 

6.  144  sq.  ft. 

7.  540  sq.  ft. 


396 


ANSWERS 


Page  345. 

8.  157.08  sq.  ft. 

9.  124.704  cu.  ft. ;   1226.25  cu.  ft.  ; 

227. 766  cu.  ft. 

10.  408.408  sq.  ft. ;  607. 133  -f  cu.  ft. 

11.  890.22  sq.  in. 

12.  381. 7044  cu.  in. 

13.  $6.14. 

14.  1385.4456  cu.  in. 

15.  50.2656  cu.  in. 

16.  1  to  3  ;  yes. 

17.  7.071  ft. 

18.  3.175425  in. 

19.  13.85  in. 

20.  50  sq.  rd. 

GENERAL  PREVIEW. 
Page  346. 


2.  $500.44. 

3.  15. 

4.  if. 


5.    J;    .5. 

7.  $3.50. 

8.  166f  ft. 


Page  347. 


9.  $696.89. 
10.  50  mi. 

12.  77;  17017. 

13.  $123.48. 

14.  23ff;  23.931. 


15.  £|. 

16.  $4.50. 

17.  $7.25. 

18.  $2.625. 

19.  14f  A. 


Page  348. 

22.  8f. 

23.  ^fj-  nearly. 

24.  12.9999984. 

25.  $30.00,  or  $31.50. 

27.  14|  cts< 

28.  $340. 

29.  263.18ff. 

30.  $352.84. 

31.  $930.23. 

32.  $359.82. 


33.  $502.42. 

34.  80. 

35.  2001.05. 

Page  349. 

36.  $100; 

37.  2520. 

38.  $15if  ; 

39.  $22.50. 

40.  G.  C.  D.  =  2  ;   L.  C.  Dd.  =r  2 

X  3  X  17  X  23  x  19  X  613. 

41.  $19.70. 

42.  $922.35. 

43.  $48.96. 

44.  8.3776  ft. 

45.  50  A. 


46 

50. 
51. 
52. 
53. 

54. 
56. 
57. 

58. 


.  192. 


Page  350. 


1039.34. 

$21.00. 

58.308. 

One  hundred  twenty-five  hun- 

dred-thousandths. 
2,  2,  3,  13,  19;  4,  6,26,38,39, 

57,247,  12,  52,  76,  78,  114, 

741,  494,  988,  1482. 


Page  351. 

62.  $281.53. 

63.  $67.20. 

64.  53.6862  -f. 

65.  2.93. 

66.  $28.80. 
68.  31|f 

70.  3.509ft. 

71.  630;  5. 

72.  420.17  times. 


ANSWERS 


397 


Page  352. 

73.  25. 

74.  4$%  ;  $288. 

75.  $344.63. 

76.  48  ft. 

77.  20091.8081. 

78.  $260.40;  $13.02. 

79.  $90.32;  $116.67;  $143.01. 

80.  $55000. 

81.  170f  yd. 

82.  56.56  -f-  rd. 

83.  150. 

84.  6  men. 

85.  $8.6625. 

Page  353. 


86.  $554.40. 

87.  $984.96. 

88.  $200;  $240 

89.  $2.00. 

90.  H- 

91.  $391.27. 

92.  $40.00. 

93.  8. 

94.  $252.53. 

95.  $36. 

96.  un- 

97.  3.21  sq.  ft. 


$280. 


Page  354. 


98.  $.005  on  $1. 

99.  5.12  ft. 

100.  $3865. 

101.  34.64  in. 

102.  $1260. 

103.  $410.84. 

104.  $27.00. 

105.  $28.80. 

106.  $152.29. 

107.  4. 

108.  $400;  $12.00. 


109.  $3.40. 

110.  680  sq.  ft.  ;  $10.88. 

Page  355- 

111.  8  men. 

112.  25%. 

113.  $1402. 

114.  $633.588. 

115.  173.57  bu. ;   138.85  bu. 

116.  56  min.  36  sec.  past  10  A.M. 

117.  10.7  ft. 

118.  43.23  in. 

119.  4  :  3. 

120.  82|  ft. 

121.  234f. 

122.  21H#- 

Page  356. 

123.  $1006.54. 

124.  10  min.  past  5  P.M.  ;  50  min. 

57.28  sec.  past  8  A.M. 

125.  2  hr.  44  min. 

126.  8.2  ft.,  very  nearly. 

128.  1  mi.  9  yd.  1  ft.  3.23  in. 

129.  133|  times. 

130.  $757.59. 

131.  $11.93. 

132.  19.6  ft.,  very  nearly. 

133.  12.15. 

134.  50.2656;  33.5104. 

135.  24  sq.  ft. ;  7.88  cu.  ft. 

Page  357. 

136.  32760. 

137.  2if7*. 

139.  6  children. 

140.  170|  min. 

141.  5  ft. 

142.  $100. 

143.  ^4:1. 

144.  68£  ft. 

145.  $200. 


398 


146.  8.2ft.,  nearly. 

147.  $0.1 5f. 

148.  25.47  ft. 

149.  165jii  lb. 

150.  2069.69  sq.  rd. 

151.  28.8  cu.  ft. 


ANSWERS 
Page  358. 


152.  12  sq.  ft.  88.56  sq.  in. 

153.  507sq.yd. 

154.  87|. 

156.  87°  33'  W. 

157.  268f  sec.  after  B 


APPENDIX. 


The  following  subjects  are  presented  in  an  appendix,  not  be- 
cause they  are  unimportant,  but  for  the  reason  that  thus  placed 
they  may  more  distinctly  constitute  a  supplementary  course  which 
the  pupil  may  elect  to  study  or  not,  as  circumstances  may  incline 
him. 

1.  Duodecimals.  9.  Circulating  Decimals. 

2.  Metric  System.  10.  G.  C.  D.  and  L.  C.  Dd.  of 

3.  Specific  Gravity.  Fractions. 

4.  Foreign  Exchange.  11.  Thermometer. 

5.  Arithmetical  Progression.      12.  The  Clock. 

6.  Geometrical  Progression.        13.  Work. 

7.  Compound  Interest  (Table).  14.  Averaging  of  Accounts. 

8.  Annuities.  15.  Miscellaneous  Exercises 

and  Problems. 

DUODECIMALS. 

Duodecimals  (Latin,  duodecim,  twelve)  are  fractions  of  which 
12  of  any  order  equal  one  of  the  next  higher  order. 

The  unit  is  the/oo£,  which  is  divided  into  12  equal  parts  called 
primes  ('),  each  prime  (')  being  divided  into  12  equal  parts  called 
seconds  (//),  each  second  (")  in  the  same  manner  into  12  thirds 
('"),  and  each  third  (/7/)  into  12  fourths  (""). 

Table. 

12  fourths  ("")  =  1  third  ('"). 

12  thirds  ('")     =  1  second  ('"). 

12  seconds  (7/)    =  1  prime  ('). 

12  primes  (')       =  1  ft. 

Hence  1'  =  ^  of  a  ft. 

1"    =  TL  of  V  =  TV  of  TV  ft.  =  ti,  of  a  ft. 

I///  =     ^  of  1"  =  TV  of  T£¥  of  1  ft.  =  T^¥  of  a  ft. 
I////  =  ^  of  V"  =  TV  of  TTVir  of  1  ft,  =  ^h*  of  a  ft. 


400  PRACTICAL  ARITHMETIC 

Duodecimals  are  employed  principally  by  artisans  in  the  meas- 
urement of  lines,  surfaces,  and  solids. 

The  adding  and  subtracting  of  duodecimals  differ  in  no  respect 
from  the  adding  and  subtracting  of  other  compound  numbers. 


EXERCISES. 

1.  What  is  the  sum  of  12  ft.  7'  10",  17  ft.  8'  9",  and  35  ft.  5'  8"? 

2.  Add  7  ft.  V  3"  6'",  1  ft.  3"  6'"  1"",  7  ft.  8'  1"  9"",  8  ft. 
10'  6". 

3.  Add  123  ft.  5'  6"  8'",  217  ft.  9'  10"  8'",  and  352  ft.  7'  9"  4'". 

4.  Add  6  ft.  4'  2",  15  ft.  4"  3'",  8'  9"  4"'  3"",  and  7"". 

5.  What  is  the  sum  of  186  ft.  5'  9"  4'"  5"",  218  ft.  7/ 10"  10"' 
8"",  and  235  ft,  6'  9"  7"'  10"". 

6.  From  9  ft.  1'  3"  take  2  ft.  6'  1"  3"'. 

7.  From  18  ft.  3"  9""  take  10  ft.  2'  2"  6"'. 

8.  From  275  ft,  5'  6"  8X"  take  127  ft.  8'  4"  5"'. 

9.  From  225  ft.  0'  2"  5'"  7""  subtract  117  ft.  5'  9"  8"'  5"". 
10.  From  a  board  15  ft.  7'  6"  in  length,  3  ft.  8' 11"  were  sawed 

off.    What  was  the  length  of  the  piece  left? 


MULTIPLICATION. 
What  is  the  product  of  8  ft.  5'  4"  and  5'  3"  ? 

Process.  Explanation. 

8ft.      5'      4"  5'  =  T52,  4"  =  Tf ¥,  3'  =  A- 

5  ft.  3'  4"  X  3'  =  Tf¥  X  A  =  THir  =  12/x/  = 

2  ft.      V      4"      Ox"        1"  O"7  ;  5X  X  3X  =  T52  X  T\  =  T1^  =  15"  ; 
42  ft.       2X      8"  15"  +  1"  =  16"  =  I/  4". 

44  ft.      ¥      0"      0/"  8  X  3X  =  8  X  &  =  f f  =  24X  ;  24X  +  1/ 

=  25X  =  2  ft.  lx. 

4"  X  5  =  Tf  ?  X  5  =  i?¥°¥  =  20"  =  V  8". 
5/  x  5  =  T\  X  5  =  ff  =  25r  ;  25r  +  \f  =  26X  =  2  ft,  2r. 
8  X  5  ==  40  ;  40  +  2  =  42  ft. 
Adding  the  two  partial  products,  we  have  44  ft.  4X  0"  0"x. 

NOTE. — '  and  ",  etc.,  are  called  indices.  Since  a  product  has  as  many 
indices  as  both  its  factors,  by  the  use  of  the  indices  the  multiplication  may 
be  performed,  in  practice,  without  the  fractions. 


APPENDIX  401 

PROBLEMS. 

1.  A  board  is  7  ft.  5'  8"  in  length  and  2  ft.  4'  7"  in  breadth. 
What  is  its  area? 

2.  How  many  cubic  feet  in  a  wall  80  ft.  9'  long,  3  ft.  ¥  high, 
and  1  ft.  Sin.  wide? 

3.  Find  the  surface  to  be  plastered  in  a  room  18  ft.  3'  long,  15  ft. 
2'  wide,  and  10  ft.  3'  high,  allowing  8'  for  the  width  of  the  base- 
board. 

4.  A  pile  of  wood  is  255  ft.  long,  4  ft.  6'  high,  and  8  ft.  4'  wide. 
Find  the  number  of  cords  in  it. 

5.  What  are  the  solid  contents  of  a  block  of  stone  3  ft.  2'  long, 
2  ft.  3'  6"  wide,  and  4  ft.  2'  high  ? 

6.  What  would  it  cost  to  plaster  a  wall  32  ft.  8'  long  and  9  ft. 
high  at  17  cents  per  square  yard? 

7.  How  many  loads  of  earth  must  be  taken  out  in  digging  a 
cellar  that  is  to  be  45  ft.  6  in.  long,  25  ft.  wide,  and  10  ft.  9  in.  deep? 


DIVISION. 
Divide  14  ft,  9'  8"  by  3  ft.  5'. 

Process.  Explanation. 

3  ft.  5'  )  14  ft.  9'  8"  ( 4  ft.  4/  14  -J-  3  =  4  +  ;  (3  ft.  5')  X  4  =  13 

13  ft.  8'  ft.  8';  subtracting  we  have  1  ft.  1'  8". 

1  ft.  1'  8"  1  ft.  1'  =  13';  13'  -s-  3  =  4'  -f  ; 

1  ft.  1'  8"  (3  ft.  50  X  4'  =  1  ft.  V  8";  subtract- 

ing we  have  0.    Hence  4  ft.  4'  is  the 
exact  quotient. 

PROBLEMS. 

1.  A  floor  contains  216  sq.  ft.  5'  10"  6X//,  and  is  10  ft.  6'  wide. 
How  long  is  it? 

2.  The  square  contents  of  a  quilt  are  14  ft.  6  in. ;  it  is  to  be  lined 
with  stuff  2  ft.  7  in.  wide.    Find  the  length  of  the  lining. 

3.  A  stick  of  timber  is  3  ft.  2  in.  wide,  and  2  ft.  9  in.  thick ;  it 
contains  176  cu.  ft.  4' 1"  6/7/.     Find  its  length. 

4.  Find  the  cost  of  carpeting  a  room  15  ft.  long,  12  ft.  wide  with 
carpet  27'  wide,  at  75  cts.  a  yard. 

26 


402  PRACTICAL   ARITHMETIC 

5.  A  plank  is  5'  thick,  20  ft.  2'  long,  and  contains  14  cu.  ft.  8'". 
How  wide  is  it? 


THE  METRIC   SYSTEM. 

The  Metric  System  is  a  decimal  system  of  weights  and  meas- 
ures ;  the  fundamental  unit  is  the  metre.  The  Standard  Metre 
is  a  bar  of  very  hard  metal,  whose  length,  as  determined  by 
French  scientists,  is  TuWiRnnr  of  a  quadrant  of  a  meridian,  and  is 
equal  to  about  39.37  inches. 

This  system  was  first  adopted  in  France  in  1795,  and  is  now 
used  also  in  Germany,  Spain,  Portugal,  Belgium,  and  Greece,  as 
well  as  in  Mexico,  Brazil,  and  most  of  the  other  States  of  South 
America.  Its  use  is  allowed  by  law  in  Great  Britain  and  in  the 
United  States,  but  as  yet  it  finds  small  favor,  except  among  scien- 
tists. 

The  principal  units  of  the  Metric  System  are  : 

1.  The  Metre  (m),  for  lengths. 

2.  The  Are  (a),  or  square  dekametre  (<idkm),  for  surfaces. 

3.  The  Stere  (st),  or  cubic  metre  (cbm),  for  volumes. 

4.  The  Litre  (*),  or  cubic  decimetre*  (cbdm),  for  small  volumes. 

5.  The  Gramme  (g),  for  weights ;  equal  to  the  weight  of  one 
cubic  centimetre*  of  water  at  4°  C.  =  39.2°  F. 

The  units  are  all  divided  and  multiplied  decimally.  Subdivi- 
sions are  indicated  by  Latin  prefixes ;  multiples,  by  Greek 
prefixes. 

(The  prefix  milli  means  Tt^  =  .001. 
The  prefix  centi  means  T^  =  .01. 
The  prefix  deci  means  TV  =  .1. 

The  prefix  deka  means  10. 
Greek    \  ^e  Prenx  hekto  means  100. 
The  prefix  kilo  means  1000. 
The  prefix  myria  means  10,000. 

Metric  numbers  are  written  decimally,  with  the  point  placed 
immediately  after  the  unit,  as  10.15™,  which  may  be  read  "  10  and 
TW  metres,"  or,  "  10  metres,  1  decimetre,  5  centimetres." 


*  1  dekametre  —  10  metres  ;  1  decimetre  =  y1^  metre  ;  1  centimetre  = 
metre. 


APPENDIX  403 

LINEAR    MEASURE. 

Table. 

10  millimetres  (mm)  =  1  centimetre  (om). 
10  centimetres  =  1  decimetre  (dm). 
10  decimetres  =  1  metre  (m). 

10  metres  =  1  dekametre  (dkm). 

10  dekametres          =  1  hektometre  (hm). 
10  hektometres        =  1  kilometre  (km). 
10  kilometres  =  1  myriametre  (Mm). 

The  metre  is  very  little  more  than  39.37  in.  The  kilometre  is  a 
little  less  than  f  of  a  mile. 

Reduction  from  one  denomination  of  the  table  to  another  is 
made  by  simply  moving  the  decimal  point  to  the  right  or  left :  to 
the  right  for  lower  denominations ;  to  the  left  for  higher  denomi- 
nations. Thus  it  will  be  seen  that  operations  with  metric  numbers 
are  similar  to  those  with  decimals. 

Illustrations. 

3568m  ==  35.68hm  =  3.568km  ;  3.568km  =  356.8dkm  =  35,680dm  =  356,800cm. 


SURFACE    MEASURE. 

Table. 

100  sq.  millimetres  (imm)  =  1  sq.  centimetre  (<icm). 
100  sq.  centimetres  =  1  sq.  decimetre  (idm). 

100  sq.  decimetres  =  1  sq.  metre  (<Jm). 

100  sq.  metres  =  1  sq.  dekametre  (<idkm). 

100  sq.  dekametres  =  1  sq.  hectometre  C*hm). 

100  sq.  hectometres          =  1  sq.  kilometre  (ikm). 

In  the  measurement  of  land  surface,  the 

Sq.  metre  =  1  ceo  tare  (ca). 

Sq.  dekametre  (idkm)  =  1  are  (a). 
Sq.  hectometre  (ihm)  =  1  hectare  (ha). 

The  are  equals  about  10|  sq.  ft.    The  hectare  equals  about 
acres. 


404  J-KACTICAL   ARITHMETIC 

Illustrations. 
1.  Change  5*hm,  3^ra,  9im,  to  sq.  metres. 

5qhm      =50,0001™ 


2.  How  many  ares  in  158a  and  3561ca? 

158a  =  158a 
3561ca  =    35.61a 
193.  61a 

MEASURE    OF    VOLUME. 
Table. 

1000  cu  millimetres  (cmm)  =  1  cu.  centimetre  C*™). 
1000  cu.  centimetres          =  1  cu.  decimetre  (cdm). 
1000  cu.  decimetres  =  1  cu.  metre  (cbm)  =  1  stere. 

The  stere  is  used  in  measuring  wood,  etc. 

10  decisteres  (ds)  =  1  stere  (st). 

10  steres  =  1  dekastere  (dkst). 

Illustrations. 

1.  Reduce  57,000ccm  to  cu.  decimetres. 

From  cu.  centimetres  to  cu.  decimetres  there  is  but  one  step  ; 
hence  57,000ccm  =  57  edm. 

2.  How  many  steres  of  wood  in  a  pile  8m  long,  1.6m  high,  and 
lm  wide? 

Volume  =  1.6m  X  8m  X  lm  =  12.8  steres. 

CAPACITY. 
Table. 

10  millilitres  (ml)  =  1  centilitre  (cl). 

10  centilitres        =  1  decilitre  (dlj. 

10  decilitres          =  1  litre  (l). 

10  litres  ==  1  dekalitre  ("»). 

10  dekalitres         =  1  hektolitre  (w). 

10  hektolitres       =  1  kilolitre  (kl)  =  1  cu.  metre. 
The  litre  is  used  in  measuring  liquids,  grain,  etc. 
The  hektolitre  is  used  in  measuring  large  quantities. 
One  millilitre  =  1  cu.  centimetre. 


APPENDIX  405 


Illustrations. 

1.  How  many  hektolitres  of  air  in  a  room  6m  long,  5m  wide,  and 
3mhigh? 

Volume  =  6m  X  5m  X  3m  =  90cbm  =  90*1  =  900hl. 

2.  If  the  value  of  a  hektolitre  of  grain  is  $1.80,  what  is  the 
value  of  a  dekalitre? 

1"  =  IQ^1 ;  therefore,  the  value  of  a  dekalitre  =  TV  of  $1.80  =  $0.18. 

WEIGHT. 
Table. 

10  milligrammes  (mg)  =  1  centigramme  (cg). 

10  centigrammes  =  1  decigramme  (dg). 

10  decigrammes  =  1  gramme  (g)  =  wt.  of  I00™  of  water. 

10  grammes  =  1  dekagramme  (***). 

10  dekagrammes  —  1  hektogramme  (h«). 

10  hektogrammes  =  1  kilogramme  (kg)  =  lodm  of  water. 

10  kilogrammes  =  1  myriagramme. 

10  myriagrammes  =  1  quintal. 

10  quintals  =  1000kg  =  1  tonneau,  or  ton  (l)  =  lcbm  of  water. 

1  cu.  decimetre  =  1  litre  ;  1  litre  of  water  =  1  kilogramme  in  wt. 

Illustration. 

Find  the  cost  of  12hg,  6dkg,  and  3dg  of  sugar,  at  the  rate  of  15  cts. 
per  kilogramme. 

12hg  =  1200* 
6dkg  _      eog 

3dg  =       0.3g 

1260.3g  =  1.2603*8  ;  1.2603  X  .15  =  18.9  cts. 

SPECIFIC    GRAVITY. 

The  Specific  Gravity  of  a  substance  is  the  ratio  of  the  weight 
of  a  given  volume  of  it  to  the  weight  of  an  equal  volume  of  water. 

Table. 

Wt.  of  1  cu.  centimetre  of  water  =  1  gramme. 
Wt.  of  1  cu.  decimetre  of  water    =  1  kilogramme. 
Wt.  of  1  cu.  metre  of  water          =  1  ton. 


40G  PRACTICAL   ARITHMETIC 

Therefore,  the  specific  gravity  of  a  substance  is  the  number 
representing  the  grammes  in  a  cu.  centimetre  of  the  substance,  the 
kilogrammes  in  a  cu.  decimetre  of  the  substance,  and  the  tons  in  a 
cu.  metre  of  the  substance. 

FORMULA. 
Specific  Gravity  =  Wt-J*L*  =  *L  =  Tons. 

com  cdm  cbm 

Volume  in  «»  =  ^Weight Jnj*_ 
Specific  Gravity 

Illustrations. 

1.  What  is  the  specific  gravity  of  a  substance  of  which  8.4ccm 
weighs  33.68? 

So  irr  =  —  =  33'6  -  4 

-  com  -    8<4    - 

2.  What  is  the  volume  of  a  body  whose  specific  gravity  is  2.5 
and  whose  weight  is  5  tons  ? 

wt       -    5   —  2cbm 
sP.  Gr.  -        -  *     ' 


.9m 
1.6km 


=  16com 

=  If cbm 
=  3fst 

=  litres 


Illustration. 

At  20  cts.  a  litre,  what  will  be  the  cost  of  150  qt.  of  olive  oil  ? 
1  liq.  qt.  =  if  1. ;  150  qt.  —  if  a  V  -U  1. ;  ^  X  ¥  =  $28.24. 


APPROXIMATE   EQUIVALENTS. 

Table. 

Metre 

=     1.1  yds. 

Yard 

Kilometre 

=        f  mi. 

Mile 

Sq.  metre 

=      1|  sq.  yd. 

Sq.  yard 

Hektare 

=      2|  acres. 

Acre 

Cu.  centimetre 

—      TV  cu.  in. 

Cu.  inch 

Cu.  metre 

=     1.3  cu.  yd. 

Cu.  yard 

Stere 

=      T3T  cord. 

Cord 

Litre 

|1TV  liq.  qt. 
(  T9o  dry  qt. 

Liq.  quart 
Dry  quart 

Hektolitre 

=      2f  bu. 

Bushel 

Gramme 

=    15|grs. 

Pound  av. 

Kilogramme 

=      2^  Ibs.  av. 

Pound  troy 

APPENDIX  407 


PROBLEMS. 

1.  Reduce  57,654m  to  millimetres ;   to  dekametres ;   to  centi- 
metres ;  to  hektometres  ;  to  decimetres  ;  to  kilometres. 

2.  Find  the  sum,  in  metres,  of  243m,  265om,  4264mm,  .012km. 

3.  Find  the  difference,  in  metres,  between  .628km  and  3158dm. 

4.  Reduce  360,000<i(im  to  sq.  metres ;  to  sq.  hektometres ;  to  sq. 
kilometres. 

5.  Find  the  cost  of  2.8ha  of  land  at  $1.05  an  are. 

6.  Find  the  ares  in  258a  and  4672ca. 

7.  Reduce  57,000ccm  to  cu.  decimetres  ;  to  cu.  metres. 

8.  How  many  cu.  decimetres  in  a  bin  measuring  12m  X  6.48m 
X  4.13m? 

9.  How  many  steres  of  wood  in  a  pile  measuring  9m  X  2.6m  x 
lm? 

10.  Find  the  hectolitres  of  air  in  a  room  measuring  5m  X  4m  X 
3m? 

11.  How  many  hektolitres  of  corn  in  a  crib  measuring  9m  X  3m 
X  2m? 

12.  How  many  litres  in  a  cistern  whose  dimensions  are  2m  X  lm 
X  .5m? 

13.  Reduce  3*  to  grammes  ;  to  milligrammes  ;  to  dekagrammes ; 
to  decigrammes. 

14.  How  many  kilogrammes  of  water  will  be  held  in  a  cistern 
that    is    cylindrical    in    shape  and  is  1.5m  in   diameter  and  4m 
deep? 

15.  From  18km  take  18mm. 

16.  How  many  metres  of  muslin,  at  $0.25  per  metre,  must  be 
given  in  exchange  for  300hl  of  oats,  at  $1.20  per  hektolitre  ? 

17.  What  is  the  area  (in  ares)  of  a  floor  3.25m  long  and  2.5m 
wide? 

18.  A  grocer  buys  butter  at  $0.28  per  lb.,  and  sells  it  at  $0.60  per 
kilogramme.    What  per  cent,  does  he  gain  or  lose? 

19.  At  39  cts.  a  metre,  what  would  it  cost  to  cut  a  ditch  9dkm  10m 
7dm  in  length  ? 

20.  How  many  metres  in  one-half  a  mile? 

21.  If  the  diameter  of  a  ball  is  63cm,  find  the  surface  and  volume 
of  the  ball  in  inches. 

22.  The  specific  gravity  of  sea-water  is  1.026,  and  that  of  milk 
1.032 ;  find  the  weight  of  a  hektolitre  of  each  in  pounds  and  in 
kilogrammes. 


408  PRACTICAL   ARITHMETIC 

23.  Find  the  volume  of  63*  of  platinum,  if  its  specific  gravity  is 
21.    Find  the  volume  also  in  eu.  inches. 

24.  If  a  pedestrian  goes  125m  in  a  minute,  what  is  his  rate  in 
miles  per  hour? 

25.  Find  the  hektares  in  a  lot  that  is  130m  square. 

26.  What  is  the  deoth  of  a  bin  12m  long  and  8m  wide,  to  hold 
2000hlof  wheat? 

27.  At  25  cts.  a  quart,  what  is  the  cost  of  101  5dl  of  oil? 

28.  How  many  metres  in  100  mi.  20  rd.  6  yd.  3  ft.  2  in.  ? 

29.  Reduce  9  myrialitres  5hl  61 10dl  to  litres. 

30.  A  vessel  full  of  alcohol,  specific  gravity  .916,  weighs  7.4kg. 
When  empty  it  weighs  500g.     How  many  litres  will  the  vessel 
hold? 


FOREIGN  EXCHANGE. 

Domestic  Exchange,  of  which  we  have  treated,  takes  place  be- 
tween different  parts  of  the  same  country.  Foreign  Exchange 
takes  place  between  different  countries. 

The  methods  of  computation  are  the  same  in  both,  except  that 
the  latter  requires  the  reduction  of  the  currency  of  one  country  to 
that  of  another. 

In  foreign  exchange  the  practice  is  to  send  three  separate  bills 
in  different  ways ;  each  bill  being  so  conditioned  that  the  payment 
of  one  of  them  cancels  the  other  two,  and  the  non-payment  of  any 
one  keeps  all  valid. 

Form  of  Draft. 


J&500  Ne^  YorK,  April  1,  1898. 

fit  sigrjt  of  tl)is  First  of  Exchange  (Second 
ar\d  Tt\ird  of  tl\e  san\e  terror1  ar^d  date  unpaid)  pay  to  tl\e  order  of 

John  Jefferson 

Five  hundred  pounds 

Yaliie  received,  ar\d  charge  to  accoiiiit  of 

To  Baring  Brothers,  BUss  $  Morton. 

London. 


APPENDIX  409 

VALUE  OF  FOREIGN  COINS. 
Proclaimed  by  Law,  July  1,  1897. 

Argentine  Republic Peso $  .965 

Austria Crown 203 

Belgium Franc 193 

Bolivia Boliviano 443 

Brazil Milreis 546 

British  Poss.,  N.  A Dollar 1.000 

Central  American  States Peso 443 

Chili Peso 365 

China,  Chefoo Tael 686 

Haikwan.   .   . Tael 730 

Shanghai     . Tael 655 

Tientsin Tael 695 

Colombia Peso .443 

Cuba Peso 926 

Denmark Crown .        .268 

Ecuador Sucre 443 

Egypt Pound 4.943 

France Franc 193 

Finland Mark 193 

German  Empire Mark 238 

Great  Britain Pound 4.866J 

Greece .  Drachma 193 

Hayti Gourde 965 

India Rupee 211 

Italy Lira 193 

Japan Yen{<*?ld "7 

(Silver 478 

Liberia Dollar 1.000 

Mexico Dollar 482 

Netherlands Florin 402 

Newfoundland Dollar 1.014 

Norway Crown 268 

Peru Sol 443 

Portugal Milreis 1.080 

Russia Rouble  (Gold) 772 

Spain Peseta 193 

Sweden Crown 268 

Switzerland Franc 193 

Turkey Piaster 044 

Venezuela Bolivar  .  .193 


410  PRACTICAL  ARITHMETIC 

Quotations. 

March  28,  1899. 
"  Sterling  Exchange,  4.85|  ®  4.86  ;  4.83}  @  4.84. 

Paris  Exchange,  5.18$  less  TV  ®  5.18| ;  5.21}  less  TV  @  5.21}." 

That  is,  in  London  the  value  of  a  pound  sterling  in  United 
States  money  varied  from  $4.85|  to  $4.86  for  sight  bills,  and  from 
$4.83}  to  $4.84  for  60-day  bills  ;  in  Paris  the  value  of  a  United  States 
dollar  varied  from  5.18'f  less  T^  fr.  to  5.18|  for  sight  bills,  and  from 
5.21}  less  TV  ct.  to  $5.21}  for  60-day  bills. 

Drafts  are  at  a  premium  or  at  a  discount,  in  accordance  with 
the  relative  condition  of  trade  between  two  countries. 

For  instance,  when  New  York  owes  London  more  than  Lon- 
don owes  New  York,  bills  on  London  are  above  par,  or  at  a 
premium ;  the  case  being  reversed,  bills  on  London  are  at  a 
discount. 

Illustrations. 

1.  What  was  the  cost  of  a  sight  draft  on  Paris  for  1000  francs, 
March  28,  1899? 

5.18|  francs  =  $1.00. 
Hence  1000  francs  =  1000  -t-  5.18f  =  $192.77,  cost. 

2.  What  was  the  cost  of  a  bill  for  £300  on  London  at  the  same 
date? 

Since  £1  =  $4.86,  £300  =  $4.86  X  300  =  $1458.00. 

PROBLEMS. 

1.  Find  the  cost  of  a  draft  on  Liverpool  for  £1000,  premium  at 
4J%.     [See  table  above  for  value  of  £1.] 

2.  On  Jan.  1,  1898,  the  interest  debt  of  Spain  was  528,185,659 
pesetas.    Find  the  value  of  this  sum  in  United  States  money? 

3.  On    the    same    date    Spain's    income    was    estimated    at 
$152,970,000.    Find  the  value  in  pesetas  and  show  the  amount  left, 
after  paying  the  interest  debt,  to  carry  on  civil  and  military 
transactions. 

4.  What  is  the  cost  of  a  bill  on  Berlin  for  2000  marks  ? 

5.  What  is  the  cost  of  a  draft  on  Glasgow  for  £500,  exchange 
being  at  par? 

6.  A  Chicago  merchant  bought  a  bill  of  exchange  on  London 
for  £795  15s.,  sterling  exchange,  as  quoted  above  for  60-day  bill. 
Find  the  cost  of  the  bill. 


APPENDIX  411 

7.  A  merchant  paid  $530  for  a  draft  on  Paris,  exchange  at 
5.18}.    Find  the  face  of  the  draft. 

8.  Find  the  cost  of  a  draft  on  Hamburg  for  13,700  marks. 

9.  If  a  Philadelphia  merchant  pays  $3058.50  for  a  bill  on  Man- 
chester, Eng.,  for  £533  15s.,  wliat  is  the  rate  of  exchange? 

10.  What  must  a  merchant  in  St.  Louis  pay  for  a  bill  on  Havre 
for  18,875  francs,  exchange  being  quoted  at  5.20f? 

11.  Find  the  cost  of  a  draft  on  London  for  £500  12s.  6d. 

12.  Find  the  cost  of  a  draft  on  Paris  for  2800  francs. 

13.  Find  the  cost  of  a  draft  on  Frankfort  for  6500  marks. 

14.  How  many  dollars  must  be  paid  in  Cairo  (Egypt)  for  a 
draft    having   a   face    value    of   £250,   exchange    being    at    \\% 
premium? 

15.  A  New  York  merchant  owing  15,000  francs  in  Paris  remits 
by  exchange  on  London.    Find  the  cost  of  his  draft  in  U.  S.  dol- 
lars, 25.22  francs  being  equal  to  £1. 


ARITHMETICAL  PROGRESSION. 

1.  An  "Arithmetical  Progression  is  a  series  of  numbers  that 
increase  or  decrease  by  a  common  difference,  as  7,  10,  13,  16,  19,  22  ; 
or,  12,  10J,  9,  7£,  6."    The  common  difference  in  the  first  series  is 
3 ;  in  the  second  it  is  \\.    The  numbers  constituting  a  series  are 
called  its  terms. 

2.  Two  principal  cases  arise :  (1.)  To  find  any  term  of  a  series; 
(2.)  To  find  the  sum  of  the  terms  of  a  series. 


1.  To  Find  Any  Term. 

Let  it  be  required  to  find  the  sixth  term,  or  any  term,  of  the 
series  7,  10,  13,.  16,  19,  etc. 

Examining  the  construction  of  the  series,  we  find  the  common 
difference  to  be  3,  and  the 

1st  term  =  7  4th  term  =  7  +  (3  X  3) 

2d  term  =  7  +  (3  X  1)  5th  term  =  7  +  (3  X  4) 

3d  term  =  7  +  (3  X  2) 

From  which  we  see  that,  to  form  the  terms  following  the  first,  the 
common  difference  was  multiplied  in  succession  by  one,  two, 
three,  and  four.  Hence  the  sixth  term  =  7  +  (3  X  5),  and  any 
term  =  7  +  (3  X  the  number  of  the  term  less  one). 


PRACTICAL  ARITHMETIC 

Were  the  series  a  decreasing  one,  the  products  above  would 
require  to  be  subtracted  instead  of  being  added,  and  any  term 
would  equal  first  term  —  (3  X  the  number  of  the  term  less  one). 

FORMULA. 
Any  term  =  1st  term  ±  (Com.  dif.)  X  (No.  of  the  term  —  1). 

NOTE. — The  sign  ±  is  read  "  plus  or  minus." 

To  apply  the  formula,  let  it  be  required  to  find  the  fortieth 
term  of  the  series  £,  f ,  1,  H,  etc. 

The  com.  dif.  =  f  —  £  =  f  —  f  =  J  ;  40th  term  =  |  +  £  (40  —  1) 

=  i  +  ¥  =  ¥  =  iQi- 

2.  To  Find  the  Sum  of  the  Terms. 

Let  it  be  required  to  find  the  sum  of  five  terms  of  the  series 
1,  3,  5,  7,  9. 

The  series,  as  given,  is    1,       3,       5,       7,       9 

The  order  reversed,  is    9,        7,       5,       3,       1 

Twice  the  sum  is  10  +  10  +  10  +  10  +  10 

or  10  X  5. 
Once  the  sum  is  1^_X_5  =  25 

10  is  the  sum  of  1  and  9,  the  first  and  last  terms  of  the  given 
series  ;  5  is  the  number  of  terms. 
Hence  the  formula : 

Sum  of  terms  =  1st  term  +  Lagttgrm  x  Number  of  terms. 

2 

To  apply  the  formula,  let  it  be  required  to  find  the  sum  of  8 
terms  of  the  series  12,  10J,  9,  etc. 

Since  we  do  not  know  the  eighth  term,  we  proceed  to  find  it 
by  the  formula  for  any  term. 

8th  term  =  12  —  (Ifc  X  7)  =  12  —  10|  =  1|. 

Sum  of  8  terms  =  12  +  *^  x  8  =  ^4  =  54. 

GEOMETRICAL  PROGRESSION. 

1.  A  Geometrical  Progression  is  a  series  of  terms  in  which  any 
term  is  equal  to  the  product  of  the  preceding  term  and  a  factor, 
which  is  constant  throughout  the  series,  as  4,  8,  16,  etc.,  whose  con- 
stant factor  is  2.  The  constant  factor  of  a  series  is  called  its 
common  ratio. 


APPENDIX  413 

2.  Two  principal  cases  arise  :  (1.)  To  find  any  term,  of  a  series; 
(2.)  To  find  the  sum  of  the  terms  of  a  series. 

1.  To  Find  Any  Term. 
Illustration. 

Let  it  be  required  to  find  the  sixth  term  or  any  higher  term  of 
the  series  2,  4,  8,  16,  etc. 

The  common  ratio  is  2,  and 

The  1st  term  =  2, 

2d  term  =  2  X  21, 
3d  term  =  2  X  22, 
4th  term  =  2  X  23, 
5th  term  =•-  2  X  2*. 

From  which  we  see  that,  to  form  the  terms  following  the  first, 
the  common  ratio  was  raised  successively  to  the  first,  second,  third, 
and  fourth  powers.  Hence,  the  sixth  term  =  2  X  25,aud  any  term 
=  2X2  raised  to  a  power  denoted  by  the  number  of  the  term  less 
one. 

Were  the  series  a  decreasing  one,  the  common  ratio  would  be  |, 

the  reciprocal  of  2. 

FORMULA. 

Any  term  =  1st  term  x  Com.  ratio  raised  to  a  power 
•whose  index  is  one  less  than  the  number  of  the  term. 

To  apply  the  formula,  let  it  be  required  to  find  the  sixth  term 
of  the  series,  5,  10,  20,  etc. 

The  com.  ratio  is  2  ;  the  6th  term  =  5X25  =  5X32  =  160. 

2.    To  Find  the  Sum  of  the  Terms. 
Illustration. 

Let  it  be  required  to  find  the  sum  of  five  terms  of  the  series  2, 
6,  18,  etc. 

The  ratio  is  3,  and  the  sum  of  five  terms  is 

2  +  6  +  18  +  54  -f  162. 
Multiplying  this  sum  by  3,  the  common  ratio,  we  have 

6  +  18  +  54  -f  162  +  486. 

Subtracting  the  upper  sum  from  the  lower,  we  have,  twice  the  sum 
=  486  —  2  ;  therefore,  once  the  sum  =  4^~^. 


414  PRACTICAL  ARITHMETIC 

But,  486  =  last  term  X  the  ratio,  while  —  2  means  "subtract 
the  first  term,"  and  2,  divisor,  =  the  ratio  —  1. 
Hence  the  formula : 

Sum  of  terms  =  Last  term  x  Ratio  —  1st  term 

Ratio  —  1 

When  the  ratio  is  less  than  1,  the  series  is  a  decreasing  one,  and 
the  formula  becomes : 

Sum  of  terms  =  1st  term  —  Last  term  X  Ratio 

1  —  Ratio 

To  apply  the  formula,  let  it  be  required  to  find  the  sum  of  3, 12, 
48,  etc.,  to  6  terms.  The  ratio  is  4. 

The  6th  term  =  3  X  45  =  3  X  1024  =  3072.    The  sum  =  3072  x  4  ~  3 

_  12288  —  3  __  12285  __  ^QQ^ 
3  3 

3.  Compound  interest  problems  may  be  solved  by  the  use  of  the 
formula  for  finding  any  term,  since  the  principal  and  the  yearly 
amounts  form  a  geometrical  series. 

Illustrations. 

1.  Find  the  amount  of  $60,  at  compound  interest  for  3  yr.,  at  5% . 
The  required  amount  will  be  60  times  the  amount  of  $1.00  for 
the  same  time. 
The  series  is 

$1,  $1  X  1.05,  $1  X  (1.05)2,  $1  X  (1.05)8. 
The  ratio  is  1.05,  and  last  term  =  $1  X  (1.05)3  =  $1.157625. 

$1.157625  X  60  =  $69.46,  Amt. 
Hence  the  formulae : 

Amount  =(14-  Rate,  raised  to  a  power 'denoted  by  the 
number  of  years)  X  Principal. 
Principal  =  Amount 


(1  -f  Rate,  raised  to  a  power  denoted  by  the 
number  of  years. 

2.  Find  what  principal  will  in  3  yr.,  compound  interest  at  6fc, 
amount  to  $1898.04. 

Principal  =  _JS^_  =  SJgBL  =  $1593.72. 

4.  Since  such  computations  as  the  foregoing  cannot  be  readily 
made,  tables  showing  the  amount  of  $1  at  compound  interest  for 
various  rates  and  lengths  of  time  are  in  common  use, 


APPENDIX 


415 


CO    O 

(-J    CO 
4^    GO 


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o-oo^jcoencn^ibo 

jJOCTiH^-^cOCOOiCO 


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O  Ci  to  CO  Cn  bO 

^^•  O^  CO  H^-  CO  Ot 

-»j  -vi  -vj  Oi  -<i  t— ' 


CO  Oi 


— 

t—  ^  CO 

CO  CO 


co  to  bo  i— '  M  H-*  o 

h-i  -<1  tO  GO  ^  O  ^4 

Oi  bO  CO  "^1  *^J  GO  )— ' 

oo  to  to  Oi  en  *vj  bO 

hS  co  Oi  cb  to  to  co 


to  to  to 

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CO  O  tO 


to  to  to  to  to 


CO  tO  CO 
•<!  GO  ht*- 


rf^^-COCObOH-'t— 'OO 

GOtOOiO4».COi^.CO^ 

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H-'t-'GOtOHJCni-'OO 


to  to  to  to  to 

O5  C7« 


GO  ^1  ^J  CO  CO 
CO  CO  CO  Oi  CO 


bO  ^l  CO  CO 


^  CO  bO  bO  I-J  H-i 
Q  £•  -<!  )-'  en  p 


COCOtOtOtOtObObObO 


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I— '   Qi   CO   ^.7 
^    O    *-    •<! 


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COCOOl— 'COOSCObO 
COCOCOOCCObOI— 'CO 
^OOOSOtbOi^'OCji 
CO  Oi  CO  bO  CO  CO  tO  CO 


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416  PRACTICAL  ARITHMETIC 

Formulae  for  the  application  of  the  above  table : 

Required  Amt.  =  Amt.  in  Table  X  Principal. 
Amt. 


Principal  = 


Time  or  Bate 


Amt.  in  Table 
Amt. 


Principal 


The  quotient,  as  found  in  the  table,  will  indicate  the  Time  or 
the  Rate,  as  the  case  may  be. 

Illustration. 
In  what  time  will  1700  dollars  amount  to  $2551.25,  at 


Time  =  T      =  TRT  =  1.50074. 

Referring  to  the  1%  column  in  the  table,  we  find  that  1.50074 
represents  6  years. 

ANNUITIES. 

1.  An  Annuity  is  a  sum  of  money,  generally  of  a  uniform 
amount,  payable  at  regular  intervals  of  time. 

2.  A  Perpetual  Annuity  continues  forever. 

3.  A  Certain  Annuity  continues  for  a  fixed  term  of  years. 

4.  A  Contingent  Annuity  depends  upon  the  "continuance 
of  some  status,  such  as  the  life  of  a  person." 

5.  An  Annuity  in  Reversion  begins  at  some  future  date. 

6.  An  Annuity  in  Arrears  remains  unpaid. 

7-  The  Final  Value  of  an  annuity  is  the  final  amount  of  an 
annuity  at  compound  interest. 

8.  The  Present  Value  of  an  annuity  will  amount  to  the  final 
value,  at  compound  interest. 

9.  To  facilitate  computation,  tables  are  customarily  used. 


FORMULA. 

Present  Value  =  Pres.  Val.  of  $1  in  Table  x  Annuity. 

Pres.  Val. 


Amount  of  An.  =  ^ 


Pres.  Val.  of  $1  in  Table 


APPENDIX 


417 


Table  II. 

SHOWING  THE  PRESENT  VALUE  OF  AN  ANNUITY  OF  $1  PER 
ANNUM,  AT  COMPOUND  INTEREST  FROM  1  YR.  TO  40  YRS.,  AT 

AND   AT  4?c. 


Yr. 

8*£ 

4# 

Yr. 

*k% 

40 

1 

0.96618 

0.96154 

21 

14.69797 

14.02916 

2 

1.89969 

1.88610 

22 

15.16713 

14.45112 

3 

2.80164 

2.77509 

23 

15.62041 

14.85684 

4 

3.63708 

3.62990 

24 

16.05837 

15.24696 

5 

4.51505 

4.45182 

25 

16.48152 

15.62208 

6 

5.32855 

5.24214 

26 

16.89035 

15.98277 

7 

6.11454 

6.00206 

27 

17.28537 

16.32959 

8 

6.87396 

6.73275 

28 

17.66702 

16.66306 

9 

7.60769 

7.43533 

29 

18.03577 

16.98372 

10 

8.31661 

8.11090 

30 

18.39205 

17.29203 

11 

9.00155 

8.76048 

31 

18.73628 

17.58849 

12 

9.66333 

9.38507 

32 

19.06887 

17.87355 

13 

10.30274 

9.98565 

33 

19.39021 

18.14765 

14 

10.92052 

10.56312 

34 

19.70068 

18.41120 

15 

11.51741 

11.11839 

35 

20.00066 

18.66461 

16 

12.09412 

11.65230 

36 

20.29049 

18.90828 

17 

12.65132 

12.16570 

37 

20.57053 

19.14258 

18 

13.18968 

12.65930 

38 

20.84109 

19.36786 

19 

13.70984 

13.13394 

39 

21.10250 

19.58449 

20 

14.21240 

13.59033 

40 

21.35507 

19.79277 

Illustrations. 
1.  What  is  the  present  value  of  an  annuity  for  $600  for  6  yrs. 


at 


Pres.  val.  =  Pres.  val.  (table)  X  An.  =  $5.242124  X  600  =  $3145.27. 

2.  A  man  49  yrs.  of  age  pays  $8686.64  for  a  life  annuity.  Reck- 
oning interest  at  3J#,  find  the  amount  of  the  annuity. 

Here  we  must  first  find  the  man's  expectancy  of  life.  Refer- 
ring to  the  Carlisle  Table,  we  find  it  to  be  somewhat  less  than 
22  years. 


Amt.  of  An.  =     s 


=  S572.73. 


27 


418 


PRACTICAL    ARITHMETIC 


Carlisle  Table  of  the  Expectancy  of  Life. 


Age. 

Expec- 
tancy. 

Age. 

Expec- 
tancy. 

Age. 

Expec- 
tancy. 

Age. 

Expec- 
tancy. 

Age. 

Expec- 
tancy. 

0 

38.72 

20 

41.46 

40 

27.61 

60 

14.34 

'80 

5.51 

1 

44.68 

21 

40.75 

41 

26.97 

61 

13.82 

81 

5.21 

2 

47.55 

22 

40.04 

42 

26.34 

62 

13.31 

82 

4.93 

3 

49.82 

23 

39.31 

43 

25.71 

63 

12.81 

83 

4.65 

4 

50.76 

24 

38.59 

44 

25.09 

64 

12.30 

84 

4.39 

5 

51.25 

25 

37.86 

45 

24.46 

65 

11.79 

85 

4.12 

6 

51.17 

26 

37.14 

46 

23.82 

66 

11.27 

86 

3.90 

7 

50.80 

27 

36.41 

47 

23.17 

67 

10.75 

87 

3.71 

8 

50.24 

28 

35.69 

48 

22.50 

68 

10.23 

88 

3.59 

9 

49.57 

29 

35.00 

49 

21.81 

69 

9.70 

89 

3.47 

10 

48.82 

30 

34.34 

50 

21.11 

70 

9.18 

90 

3.28 

11 

48.04 

31 

33.68 

51 

20.39 

71 

8.65 

91 

3.26 

12 

47.27 

32 

33.03 

52 

19.68 

72 

8.16 

92 

3.37 

13 

46.51 

33 

32.36 

53 

18.97 

73 

7.72 

93 

3.48 

14 

45.75 

34 

31.68 

54 

18.28 

74 

7.33 

94 

3.53 

15 

45.00 

35 

31.00 

55 

17.58 

75 

7.01 

95 

3.53 

16 

44.27 

36 

30.32 

56 

16.89 

76 

6.69 

96 

3.46 

17 

43.57 

37 

29.64 

57 

16.21 

77 

6.40 

97 

3.28 

18 

42.87 

38 

28.96 

58 

15.55 

78 

6.12 

98 

3.07 

19 

42.17 

39 

28.28 

59 

14.92 

79 

5.80 

99 

2.77 

EXERCISES   AND   PROBLEMS. 

1.  Find  the  twelfth  term  of  3,  6,  9,  etc. 

2.  Find  the  twentieth  term  of  1,  8,  15,  22,  etc. 

3.  Find  the  seventh  term  of  99,  92,  85,  etc. 

4.  Find  the  twenty-fifth  term  of  100,  96,  92,  etc. 

5.  Find  the  number  of  strokes  made  in  a  day  by  a  clock  strik- 
ing the  hours  only. 

6.  The  first  term  of  an  arithmetical  progression  is  7,  the  last 
term  79,  and  the  number  of  terms  15.    What  is  the  sum  of  the 
series  ? 

7.  The  series  is  £,  f ,  1,  etc.    Find  the  one-hundredth  term  and 
the  sum  of  the  series. 

8.  A  body  falling  for  12  sec.  passes  through  16^  ft.  the  first 
second,  and  increases  its  speed  32£  ft.  each  succeeding  second. 
What  is  the  whole  extent  of  its  fall  ? 

9.  100  apples  were  placed  in  a  row  2m  apart,  and  a  basket  was 
placed  2m  from  the  first  apple.    A  boy,  starting  at  the  basket, 
brought  to  the  basket  the  first  apple,  next  the  second  apple,  and 


APPENDIX  419 

so  on  until  all  were  brought.     How  far  did  the  boy  walk  in  per- 
forming the  labor? 

10.  A  debt  can  be  discharged  in  a  year  by  paying  $1  the  first 
week,  $3  the  second  week,  $5  the  third  week,  and  so  on.    Find  the 
last  payment  and  the  amount  of  the  debt. 

11.  Find  the  sixth  term  of  the  series  2,  6,  18,  etc. 

12.  Find  the  sixth  term  of  the  series  8,  4,  2,  etc. 

13.  Find  the  seventh  term  of  the  series  f ,  |,  |,  etc. 

14.  Find  the  nineteenth  term  of  the  series  4,  7,  10,  etc. 

15.  In  going  a  9-days'  journey  a  man  travelled  30*™  the  first 
day,  and  constantly  thereafter  increased  his  daily  distance  10km. 
How   far   did   he    travel    the    last   day,    and   how   many   miles 
altogether  ? 

16.  Starting  with  a  man's  immediate  parents  and  running  back 
10  generations,  compute  the  number  of  his  ancestors. 

17.  What  is  the  amount  of  $1  at  compound  interest  for  6  yrs.  at 
7^?    Solve  in  three  ways. 

18.  What  is  the  amount  of  $2000  for  9  yrs.  at  $%  ?    Solve  by 
the  table. 

19.  What  principal,  at  compound  interest  for  5  yrs.  at  6ft,  will 
amount  to  $267,646? 

20.  In  what  time  will  the  same  principal,  at  5%,  amount  to  the 
same  sum  ? 

21.  A  man  23  yrs.  old  has  a  life  annuity  of  $700.    Find  its 
present  value  at  Z\%. 

22.  A  woman  36  yrs.  old  has  a  life  annuity  of  $1200.    Find  its 
present  value  at  4%. 

23.  A  person  76  yrs.  old  has  a  life  annuity  of  $3000.    Find  its 
present  value  at  Z\%. 

24.  A  dowager  50  yrs.  old  has  a  jointure  of  $4500.    Find  its 
present  value,  interest  at  1%. 

25.  A  widow  29  yrs.  old  has  a  dower  of  $1800.     Find  its  present 
value,  interest  at  Z\%. 

26.  A  man  35  yrs.  old  pays  $9368.10  for  a  life  annuity.    Find  the 
amount  of  the  annuity,  interest  at  ?>\%. 

27.  A  life  annuity  costs  a  person  44  yrs.  old  $5933.35.     Find  the 
amount  of  the  annuity,  interest  at  3?%. 

28.  Find  the  amount  of  $365  at  compound  interest  for  20  yrs. 
at  5$. 

29.  Find  the  amount  of  $1728  at  compound  interest  for  25  yrs. 


420  PRACTICAL  ARITHMETIC 


CIRCULATING  DECIMALS. 

|  =  ~  =  .4,  quotient  exact,     f  =  s-~  =  .4285  -f,  quotient  inexact. 

To  reduce  a  common  fraction  to  a  decimal,  we  annex  ciphers  to 
the  numerator,  perform  the  operation  indicated,  and  point  off  in 
the  quotient  as  many  places  for  decimals  as  there  are  ciphers  an- 
nexed. Annexing  a  cipher  to  the  numerator  multiplies  it  by  the 
factors  2  and  5 ;  hence,  when  the  denominator  contains  no  other 
factors  than  2  and  5,  the  quotient  will  be  exact ;  when  it  contains 
other  factors  than  2  and  5,  the  quotient  will  be  inexact. 

T\  reduced  to  a  decimal  becomes  .384615,  and  so  on  without 
end  ;  but  in  this  instance  a  very  noticeable  peculiarity  is  that  the 
384,615  will  be  constantly  repeated,  however  far  the  reduction  be 
carried. 

Such  a  decimal  fraction  is  called  a  circulating  or  repeating 
decimal.  To  show  the  fact  of  repetition  we  place  a  point  over 
the  first  and  the  last  digit,  thus :  384615.  T3T  =  .272727,  etc.  =  .27. 

The  constantly  repeating  figures  are  called  a  repetend.  A 
mixed  repetend  begins  with  one  or  more  non-repeating  figures,  as 

.5243. 

The  practical  advantage  of  recognizing  circulates  will  appear 
in  the  following  illustrations  : 

1.  Reduce  .72  to  a  common  fraction. 


100  times  .72  =  72.7272 

1  time    .72  =      .7272 

99  times  .72  =  72 
1  time   .72  =  £f  =  T8T. 

2.  Reduce  .1172  to  a  common  fraction. 

.1172  =  .ll|f  =  .11T8T  =  ^  =  T¥oV 
Hence  the  formula : 

_Digits_of_repetend^__  =  value  of  repetencl. 
As  many  9's  as  digits 


421 


EXAMPLES. 
Reduce  the  following  repetends  to  common  fractions : 


1.  .3. 

7.  .753. 

13.  .852. 

19.    .0003. 

25.     .2297. 

2.  .4. 

8.  .216. 

14.  .144. 

20.     .246789. 

26.  2.1873. 

3.  .6. 

9.  .531. 

15.  .527. 

21.     .2564. 

27.    .4306. 

4.  .36, 

10.  .0234. 

16.  .0009. 

22.     .8716. 

28.  5.0415. 

5.  .21. 

11.  .8232. 

17.  .048. 

23.    .35135. 

29.     .84234. 

6.  .018. 

12.  .81. 

18    .08199. 

24.  3.04i2. 

30.  4.2674. 

THE   GREATEST   COMMON  DIVISOR  OF 
FRACTIONS. 

Find  the  G.  CD.  of  *|,  f,  f 

Each  of  the  given  fractions  divided  by  the  G.  C.  D.  must  give 
an  integer  for  quotient.  In  dividing  by  a  fraction,  we  invert  the 
divisor  and  then  multiply.  Hence  the  G.  C.  D.  sought  must  have 
for  its  numerator  the  G.  C.  D.  of  the  given  numerators  and  the 
L.  C.  Dd.  of  the  given  denominators. 

The  G.  C.  D.  of  2,  4,  and  6  =  2. 
The  L.  C.  Dd.  of  3,  5,  and  7  =  105. 
Hence  the  G.  C.  D.  of  f ,  f ,  and  f  =  rf  5. 


GK  C.  D.  = 


FORMULA. 

G-.  C.  D.  of  Numerators 
L.  O.  Dd.  of  Denominators' 


THE  LEAST   COMMON  DIVIDEND   OF 
FRACTIONS. 

Find  the  L.  C.  Dd.  of  f ,  f ,  and  TV 

TL«  L.  C.  Dd.  sought,  when  divided  by  each  of  the  given  frac- 
tions, must  give  an  integral  quotient.  In  dividing  by  a  fraction, 
we  invert  the  divisor  and  then  multiply.  Hence  the  L.  C.  Dd. 


422  PRACTICAL   ARITHMETIC 

required  must  have  for  its  numerator  the  L.  C.  Dd.  of  the  given 
numerators  and  the  G.  C.  D.  of  the  denominators. 

The  L.  C.  Dd.  of  2,  4,  and  8  =  8. 
The  G.  C.  D.  of  3,  9,  and  15  «=  3. 

Hence  the  L.  C.  Dd.  of  f ,  f ,  and  T8T  =  f . 

FORMULA. 

L  C  Dd  —  L-  G-  Pd-  of  Numerators 
~  G.  C.  D.  of  Denominators' 

EXERCISES. 

1.  Find  the  G.  C.  D.  of : 

1.  f,  f ,  f .  3.  12i,  3|,  17}.  5.  1A,  1&,  1&. 

*•  i7*,  A,  A,  A-     4-  10,  2j,  f .       .      6.  if,  H,  If 

2.  Find  the  L.  C.  Dd.  of: 

1.  f,  TV,  f .  3.  6f ,  3f ,  2f.  5.  A,  rV,  2i  5,  6}. 

2.  I,  if,  if.  4.  ff,  A,  &.  6.  H»  f  f »  If 

THE  THERMOMETER. 

The  Thermometer  is  an  instrument  for  measuring  change  of 
temperature  by  means  of  the  expansion  of  liquid  substances. 
Mercury  is  the  pre-eminently  suitable  substance. 

Thermometers  are  of  three  principal  kinds :  The  Fahrenheit, 
the  Centigrade,  and  the  Reaumur.  The  Centigrade  is  used  largely 
for  scientific  purposes. 

It  is  sometimes  necessary  to  transform  readings  from  one  scale 
to  another. 

Freezing  Point.    Boiling  Point.       Interspace. 

Fahrenheit 32°  212°  180° 

Centigrade 0°  100°  100° 

Reaumur 0°  80°  80° 

Hence  the  number  of  degrees  F.  —  32°  =  \\%  or  f  C.  =  Vo°-  or 
|B. 

From  which  we  deduce  the  formulae  : 

1.  (F.  —  32°)  X  f  =  C.  3.   O.  X  |  =  F.  —  32°. 

2.  (F.  —  32°)  X  $  =  R.  4.   R.  X  f  =  F.  —  32°. 


APPENDIX  423 

1.  Change  40°  F.  to  C. 

40°  —  32°  =  8°.     8°  X  f  =  -V-°  =  4f°  C. 

2.  Change  40°  C.  to  F. 

40°  X  |  =  *P  =  72°.    72°  +  32°  =  104°  F. 
The  minus  sign  ( — )  prefixed  to  a  reading  signifies  below  zero. 

3.  Change  —10°  C.  to  F. 

—  10°  X  f  =  =nr°  =  — 18° 

We  now  have  —18°  +  32°.  Difference  of  sign  implies  subtrac- 
tion. —18°  -f  32°  =  14°  F.  above  zero. 

4.  Change  —30°  C.  to  F. 

-  30°  X  f  =  =^0  =  —  54°. 
—  54°  -f  32°  =  —  22°  F.;  that  is,  22°  below  zero. 

5.  Change  32°  F.  to  C.  and  B. 

6.  Change  50°  C.  to  F.  and  B. 

7.  Change  —20°  C.  to  F.  and  B. 

THE    CLOCK. 

The  hour  and  minute  hands  are  together  at  12  o'clock,  and  the 
minute  hand  may  be  regarded  as  setting  out  at  that  point  to  rejoin 
the  hour  hand.  In  60  minutes  the  minute  hand  will  have  re- 
turned to  12,  but  the  hour  hand  will  have  passed  on  to  1.  In  60 
minutes,  therefore,  the  minute  hand  has  gained  on  the  hour  hand 
11  spaces.  Hence,  a  single  space  was  gained  in  ^  of  60  minutes  = 
5T5T  minutes. 

1.  At  what  time  between  4  and  5  o'clock  are  the  hands  of  a 
clock  together? 

At  4  o'clock  the  hands  are  4  spaces  apart.  Since  the  minute 
hand  gains  1  space  in  5T5r  minutes,  it  will  gain  4  spaces  in  4  times 
5^r  minutes  =  21T9T  minutes.  Hence,  the  hands  will  be  together 
at  21T9T  minutes  past  4. 

2.  When  will  the  hands  of  a  clock  be  together  between  : 

1.  6  and  7  o'clock  ?  4.  3  and  4  o'clock  ? 

2.  8  and  9  o'clock?  5.  9  and  10  o'clock. 

3.  1  and  2  o'clock?  6.  5  and  6  o'clock. 


424  PRACTICAL  ARITHMETIC 

3.  When  will  the  hands  of  a  clock  be  opposite  each  other 
between  : 

1.  12  and  1  o'clock?  3.  9  and  10  o'clock? 

2.  3  and  4  o'clock?  4.  11  and  12  o'clock? 

4.  When  will  the  hands  of  a  clock  be  at  right  angles  between  : 

1.  2  and  3  o'clock  ?  3.  9  and  10  o'clock  ? 

2.  4  and  5  o'clock?  4.  11  and  12  o'clock? 

5.  When  between  those  hours  will  they  make  with  each  other 
an  angle  of  30°  ?    Of  60°  ?    Of  120°  ?    Of  150°  ? 

WORK. 

1.  A.  can  do  a  certain  piece  of  work  in  6  days,  B.  in  8  days,  and 
C.  in  9  days.    How  long  will  it  take  them  to  do  it  together? 

A.  can  do  £  of  the  work  in  1  day. 

B.  can  do  |  of  the  work  in  1  day. 

C.  can  do  ^  of  the  work  in  1  day. 
All  can  do  i  +  i  -f  £  =  f  f  in  1  day. 

All  can  do  -fa  of  the  work  in  ^  of  1  day. 
All  can  do  ff  of  the  work  in  |f  =  2£f  days. 

2.  A.  and  B.  together  can  do  a  piece  of  work  in  2}  days,  A.  and 
C.  in  3J  days,  B.  and  C.  in  3|  days.    How  long  will  it  take  the 
three  working  together  to  do  the  work,  and  how  long  will  it  take 
each  alone  ? 

A.  and  B.  in  one  day  can  do  ~  °f  the  work. 

A.  and  C.  in  one  day  can  do  •—  of  the  work. 

B.  and  C.  in  one  day  can  do  ~  of  the  work. 

2A.  +  2B.  +  2C.  in  one  day  can  do  ~  +  -^  +  ~  of  the  work. 
A.  +  B.  -f  C.  in  one  day  can  do  £  +  ~-  +  -J-  of  the  work  =  I  -f 

"73  1/2 

&  +  &  =  ^-^  =  f*  of  the  work. 

In  one  day  A.  can  do  |  f  —  ~  =  f  f  —  $$  =  ft  of  the  work. 

Hence  A.  can  do  f§  in  ff  or  4T\  days. 
Find  time  required  by  B.  and  C. 

3.  If  it  takes  A.,  working  alone,  4  days,  B.  3  days,  and  C.  4£ 
days  to  do  a  piece  of  work,  how  long  will  it  take  them  to  do  the 
work  if  all  three  work  together? 


APPENDIX 


425 


4.  A.  can  do  a  piece  of  work  in  10  days,  A.  and  C.  can  do  it  in 
7  days,  and  A.  and  B.  can  do  it  in  6  days.     How  long  will  it  take 
them  all  to  do  it? 

5.  One  pipe  can  fill  a  cistern  half  full  in  f  of  an  hour,  and  an- 
other can  fill  it  three-quarters  full  in  J  an  hour.    How  long  will  it 
take  both  pipes  together  to  till  the  cistern  ? 

6.  Pipes  A.  and  B.  can  fill  a  cistern  in  3  minutes  and  5  minutes 
respectively,  and  C.  can  empty  it  in  7J  min.    In  what  time  will 
the  cistern  be  filled  when  A.,  B.,  and  C.  are  all  open  ? 

7.  A.,  B.,  and  C.  together  can  do  a  piece  of  work  in  10  days, 
A.  and  B.  together  in  12  days,  B.  and  C.  together  in  20  days.    How 
long  will  it  take  each  alone  to  do  the  work? 

8.  A.  does  T4T  of  a  piece  of  work  in  6  days,  when  B.  comes  along 
and  helps  him,  and  they  finish  it  in  5  days.    How  long  would  it 
take  B.  alone  to  do  the  work  ? 

9.  A.  and  B.  can  do  a  piece  of  work  in  4  days,  A.  and  C.  in  6 
days,  and  A.,  B.,  and  C.  in  3  days.     In  how  many  days  can  each 
do  the  work  alone? 

10.  A  reservoir  has  two  sluices,  one  of  which  alone  would  drain 
it  in  7  hours  and  the  other  in  13  hours.  How  soon  would  it  be 
emptied  if  both  were  opened  together? 

AVERAGING  OP  ACCOUNTS. 

1.  Find  the  average  term  of  credit  of  the  following  account : 
DR.  JACOB  HART.  CR. 


1898 

1 

1898 

Jan.    1 

To  Mdse 

448 

00 

Jan.  20 

Amt.  br.  forwd 

560 

00 

Feb.    4 

"   Cash 

364 

00 

Feb.  11 

By  1  Carriage 

264 

00 

"     20 

u        u 

232 

00 

"    25 

"   Cash 

900 

00 

Process. 


DR. 


Due. 


Da.    Items.       Prod. 


Jan.    1,        00 

Feb.    4,        34 

"      20,        50 


448  00000 
364  12376 
232  11600 


Due. 
Jan.  20, 
Feb.  16, 

"     25, 


1044        23976 


CR. 

Da.  Items. 

Prod. 

19   560 

10640 

46   264 

12144 

55   900 

49500 

1724 

72284 

1044 

23974 

oces,  680 

'48308 

48308  -=-  680  =  71  da.    Jan. 1  +  71  da.  =  March  13. 


426 


PRACTICAL  ARITHMETIC 


Explanation. 

Jan.  1,  the  earliest  date,  was  assumed  to  be  the  starting  point  or  focal 
date.  From  Jan.  1  to  Jan.  1  there  are  0  days  ;  from  Jan.  1  to  Feb.  4  there 
are  34  days ;  from  Jan.  1  to  Feb.  20  there  are  50  days.  On  the  credit  side 
we  proceed  in  a  similar  way,  saying  from  Jan.  1  to  Jan.  20  are  19  days, 
and  so  on.  The  balance  of  the  products  divided  by  the  balance  of  the 
items  gives  the  average  term  of  credit,  71  da.,  which,  added  to  Jan.  1, 
gives  us  March  13  as  the  day  of  payment. 


2.  When  should  interest  begin  on  the  following  account : 


DR. 


JACOB  JOHNSON. 


CR. 


1898 

1898 

Jan.    1 

To  Mdse,  3  mo. 

145 

86 

May  11 

By  Cash 

11 

00 

"      12 

u         u         5     ii 

37 

48 

July  12 

u      u 

15 

00 

June  3 

11             It              0       U 

12 

25 

Oct.  12 

u      u 

82 

00 

Aug.  4 

II             II              O       II 

66 

48 

Process. 


DR. 


CR. 


Due.         Da.     Items.  Prod. 

April  1,      00    145.86  0000.00 

June  12,      72      37.48  2698.56 

Sept.   3,     155      12.25  1898.75 

Oct.     4,     186      66.48  12365.28 

262.07  16962.59 

108.00 
Bal.  of  Items,  154.07 


Due.  Da.  Items.  Prod. 

May  11,  40  11.00  440.00 

July  12,  102  15.00  1530.00 

Oct.  12,  194  82.00  15908.00 


108.00    17878.00 

16962.55 

Bal.  of  Prod.,    915.41 


915  •+•  154.07  =  6  da.    April  1  —  6  da.  =  March  26,  1898. 


Jan.  1  +  3  mo.  —  April  1 
Jan.  12  +  5  mo.  =  June  12 
June  3  +  3  mo.  =  Sept.  3 
Aug.  4  +  2  mo.  =  Oct.  4 


which  shows  April  1  to  be  the 
focal  date.  We  now  proceed  as 
before. 


It  must  be  observed  that  when  the  balances  are  both  on  the  same  side 
of  the  account,  the  term  of  credit  must  be  added  to  the  focal  date ;  other- 
wise subtracted. 


APPENDIX 


427 


3.  Find  the  average  term  of  credit  of  the  following  account : 
DR.    RICHARD  STEVENS,  in  Acct.  with  HENRY  BECK.    CR. 


1898 

Apr.  10 

To  Mdse 

150 

00 

1898 

Apr.  30 

By  Cash 

250 

00 

"'  30 

u         u 

400 

00 

May  1 

u         u 

200 

00 

May  16 

u         « 

100 

00 

Jun.  27 

((             U 

400 

00 

Jun.  24 

«         <( 

500 

00 

4.  What  will  be  the  cash  balance  of  the  following  account  Jan. 

1,  1899,  interest  at  6f0  ? 

DR.                             ENOCH  HOBSON.                            CR. 

1898 

July  10 

To  Mdse,  2  mo. 

500 

00 

1898 

July  20 

By  Cash 

400 

00 

Aug.  1 

"       "       3    " 

700 

00 

Aug.20 

u         « 

1000 

00 

Sept.  9 

((             U              ^        (( 

800 

00 

"    20 

<(             It              O       H 

600 

00 

REVIEW. 


Simplify  the  following : 


I  X  |  X 


3. 


5 


*f  +  5* 


26.7  —  11.80  +  6.45 


f  X  3H  X  .72 
6   3f  -  f  X  4.2 


7. 


4f  X  A  +  1.8 


.25  X 


n.it=ii 

4fX5f 

12  3j  X  2J  If 

4  21  +  1J' 

13  f  X  1.25 

'  5f  —  4.25' 

14  8$  X  1TV  +  4rV  —  3ft 

6*  -  7f  -3-  28  A  +  i  ' 

15.  7^  X  T7 

H-i 

16.  4J  of  1  A-  of  A  +  31  +  2 J  of  If 

1  7       3f    Of    1-jrV 

JL  /  • 


!  of  6f 


18. 


19. 


2|  of  WT  +  yV  of 


10. 


+  I  X  H  +  3  + 


2|  of  IT>T  — 

20    ^  +  5^  of  f 
'  t  of  5f  +  f 


428  PRACTICAL  ARITHMETIC 

21.  '4    ,   2|    ,    2 1    ,    _2f 


23. 


f  of  f 


of  « 


5TV-4M        7if-6i 

24.  tt=|* 

25.  SjjV  +  2f  —  3 1  +  6|  +  lOf . 


28.  (2*  + 


(2J  —  11). 


If  of 


2f  of  3f 


-  4    of 


-s-4*. 


33.  %  —  * 
34.|^|f 
35.  JL* 


-4* 


36. 


.04478256  ^-  5.48 
.036X2.043 

6?  — 


37.  If  of  If  + 
38. 


2|. 


41.  3.01  —  5.314  +  2.4. 

42.  |-  + 2^1 +  £!  —  •&  +  ? 

43  15!  — 4f  of  If 
'  |  of  231  +  2if 

44  -02048 

'  .00003125' 

45«  TT  -*-  12f  +  A  ()f  9|. 


5i-4| 


-f-  1|  Of  If 


46. 


48.  .111  +  .  6666|  +.222222$. 


so.  4  +  _j, 


51. 


53. 


1 
HX  2^  +  21x5^ 


~l|      2H 


3*  of  41 


(2J  —  i)  of  (3i  - 


54.  -321  X. 321 -. 179  X. 179 
.321  —  .179 

„  .562  X. 562  — .188  X. 188  , 
55-  -    .562 -.188   -°f$75 


57  (-OQ056542)2 
(12.534)2  * 


40.  3J  +  2|  of  5J  of  1¥^  _  2TV  X  f  - 


APPENDIX 


429 


, 
• 


If  of  If 


62 


T%  of   1.82  +  |  of  .35   , 
— 


_  . 

-4^ 

65.  of  2  yd.  2  ft.  11J  in. 
6i 

2   of  4 

66.  (f  of  f  +  f  of  2i)  X  - 


3  + 


X  3i 


70 
}> 


7i.  «  .  _J_- 

T.5-f 


~  10 


79.          of 


. 


J> 


FTTTT+l+l 


2|_3 

—  2TT7  I  +  -S  — 


.0038425  —  .00183 


of  2.179  -  |  of  .8684 
i 


Q, 


86> 


87.  -.  X  of  12s.  9fd. 

2.1742       2.78 


.203  X  .0003  X  16 
.008  X  .0029 


430  PRACTICAL  ARITHMETIC 


93    >   X  X  *  X  *  X 

''**        2*3*5*4 

94.  .016  +  4.0808  —  .0008  +  50.1  —  .1966. 


5 


of  25r  +  j  of  1?)  . 


AX  If  XI}-  rV      '   I  of  3-f7X 

96.  A  of  **~~k  of  (3f  +  i  -  2^). 


97.  i  of  (i  +  i  +i)  +  7  X  GV  +  A)  -  A  •* 

98.  If  of  5T4T  —  1/3  of  5f  -f  Iff  of  2TV 

99.  |  (!  X  i  +  |  -f-  f)  -  ||  of  ^  +  f  of  ^. 
100.  l/  7.4538  —  6.8  -5-  8.5  —  2.03  X  1.17. 


MISCELLANEOUS   PROBLEMS. 

1.  Prove  the  product  will  be  the  same  in  whatever  order  the 
factors  be  taken. 

2.  Prove  that  in  division  of  fractions,  multiplying  the  divi- 
dend by  the  divisor  inverted  will  produce  the  quotient. 

3.  Show  how  you  determine  whether  a  given  common  frac- 
tion can  be  exactly  expressed  as  a  decimal,  and  give  reasons. 

4.  What  factors  of  two  or  more  numbers  must  be  combined  to 
produce  their  greatest  common  divisor,  and  what  ones  to  produce 
the  least  common  dividend. 

5.  State  a  method  of  multiplying  a  fraction  by  a  fraction,  and 
demonstrate  the  correctness  of  the  method. 

6.  On  what  theory  was  the  length  of  the  metre  originally  de- 
termined ? 

7.  Explain  the  process  of  finding  the  greatest  common  divisor 
by  division. 

8.  Prove  that  any  common  divisor  of  two  numbers  is  a  divisor 
of  their  sum  and  of  their  difference. 

9.  Explain  a  method  of  finding  the  greatest  common  divisor 
of  two  fractions. 

1O.  Given  interest,  principal,  and  time,  how  may  the  rate  be 
found  ? 


APPENDIX    r  431 

11.  Explain  and  illustrate  a  method  of  finding  the  least  com- 
mon dividend  of  fractions. 

12.  Indicate  the  following  operations  by  signs  in  one  connected 
expression  :  The  sum  of  3  and  4  multiplied  by  the  difference  be- 
tween 9  and  5,  and  the  product  divided  by  2  times  7.    Perform  the 
operations  indicated. 

13.  Write  a  complex  fraction.    State  the  reasons  for  regarding 
it  as  complex.    Reduce  it  to  a  simple  fraction,  and  this  result  to  a 
decimal. 

14.  Distinguish  between  a  compound  and  a  denominate  num- 
ber ;  also,  interest  and  discount.    Illustrate  by  examples. 

15.  Write  a  number  which  shall  be  at  the  same  time  simple, 
composite,  abstract,  and  even.     State  why  it  fills  each  of  these 
requirements. 

16.  Name  the  principal  unit  of  length,  of  surface,  of  capacity, 
and  of  weight  in  the  metric  system,  and  show  the  relation  among 
these  units. 

17.  The  sum  of  two  numbers  is  260  and  their  difference  is  12 ; 
find  the  numbers  and  demonstrate  the  principle  involved. 

18.  A.  and  B.  can  do  a  piece  of  work  in  three  days  ;  B.  and  C. 
can  do  it  in  four  days ;  A.  and  C.  can  do  it  in  six  days  ;  if  all  work 
together  for  the  same  length  of  time,  what  part  of  the  sum  paid 
to  all  should  each  receive? 

19.  Find  the  fourth  term  of  the  following  proportion,  and 
demonstrate  the  principle  on  which  it  is  based :  8  :  12  =  10  :  x. 

20.  A  cylindric  vessel  is  8  ft.  in  diameter ;  how  deep  must  it 
be  to  contain  75  bbl.  of  water? 

21.  Find  the  square  root  of  104976,  and  give  a  reason  for  each 
step  in  the  process. 

22.  Deduce  a  rule  for  finding  the  sum  of  an  arithmetic  series, 
and  illustrate  its  use  by  finding  the  sum  of  ten  terms  of  the  series 
whose  first  term  is  2  and  whose  common  difference  is  4. 

23.  Find  the  sixth  root  of  191102976,  and  show  why  you  believe 
your  method  to  be  correct. 

24.  Indicate  the  following  by  signs  :  The  difference  of  9  and  5 
is  multiplied  by  8,  this  product  is  divided  by  10  and  the  quotient 
increased  by  1,  the  sum  is  squared,  increased  by  2,  and  the  cube 
root  of  the  result  taken. 

25.  A  franc  is  worth  9.5d.  ;  a  mark  is  worth  11.7d. ;  a  pound 
sterling  is  worth  $4.86.    Find  the  value  of  $100  in  each  of  the  three 
other  currencies, 

3 


432  PKACTICAL  ARITHMETIC 

26.  Explain  and  illustrate  a  method  of  finding  the  least  com- 
mon denominator  of  fractions. 

27.  Insert  four  geometric  means  between  3  and  96.    Insert  two 
arithmetic  means  between  3  and  96. 

28.  Assuming  that  iron  is  7.8  times  as  heavy  as  water,  find  the 
weight  in  kilogrammes  of  a  round  bar  of  iron  .60  centimetres  in 
diameter  and  3  metres  long. 

29.  Find  the  cost  in  United  States  money  of  a  bill  of  exchange 
on  London  for  £12  15s.  9d.,  exchange  being  at  $4.86. 

30.  A  room  5m  long,  4m  wide,  and  3m  high  has  opening  from 
it  one  door  2m  high,  l^m  wide,  and  two  windows,  each  2J[m  high, 
lm  wide.    Find  the  cost  of  plastering  the  walls  and  ceiling  at  15 
cts.  a  square  metre,  deducting  half  the  openings. 

31.  In  an  arithmetic  progression  of  8  terms,  the  first  term  is  3 
and  the  last  is  31 ;  find  the  remaining  terms. 

32.  Show  the  exact  value  of  the  decimal  .666  ....  to  infinity. 

33.  The  length  of  a  tank  which  holds  100  bbl.  of  water  is  twice 
its  height,  and  its  height  is  twice  its  width  ;  find  its  dimensions 
to  the  nearest  inch. 

34.  Loaned  $6000  to  be  paid,  with  interest  at  6%,  in  six  equal 
annual  instalments  ;  what  is  the  amount  of  each  payment? 

35.  Explain  a  method  of  finding  difference  of  longitude  from 
difference  of  time,  and  show  its  application  in  finding  the  longi- 
tude of  a  place. 

36.  A  bar  of  aluminum  2cm  thick  and  2cm  wide  weighs  l£k«  ;  find 
its  length,  assuming  that  aluminum  is  2£  times  as  heavy  as  water. 

37.  State  the  process  of  finding  the  cost  in  U.  S.  money  of  a 
time  draft  on  a  foreign  country,  giving  the  reasons  for  each  step. 

38.  Reduce  the  repetend  .16213  to  a  common  fraction  in  its 
lowest  terms. 

39.  The  diameter  of  a  cylindric  vessel  is  42cm  and  its  depth  is 
6£dm  ;  how  many  litres  of  water  will  it  hold  and  how  many  kilos 
will  this  water  weigh  ? 

40.  What  factors  of  two  or  more  numbers  must  be  combined 
to  produce  their  greatest  common  divisor,  and  what  ones  to  pro- 
duce the  least  common  dividend?    Give  reasons. 

41.  Find  the  G.  C.  D.  and  the  L-  C.  Dd.  of  |,  f,  f.    Explain  the 
process  fully. 

42.  When  it  is  Monday,  7  A.M.,  at  San  Francisco,  longitude 
122°  24' 15"  W.,  what  day  and  time  of  day  is  it  at  Berlin,  longitude 
13°  23' 55"  E.? 


APPENDIX  433 

43.  A  gallon  contains  231  cu.  in.  ;  a  cubic  foot  of  water  weighs 
62.5  Ib.  ;  mercury  is  13.5  times  as  heavy  as  water.     How  many 
gallons  of  mercury  will  weigh  a  ton  ? 

44.  Find  the  sum  of  23.3,  42.61,  78.3452. 


\  2  —  *\ 


45.  Find  the  value  of  \  2  —  *\2  +      .8. 

46.  At  how  many  minutes  after  3  o'clock  will  the  hour  and 
minute  hands  of  a  watch  be  opposite  each  other? 

47.  A  general  formed  his  army  into  a  solid  square,  and  had 
200  men  left  over  ;  he  then  received  a  reinforcement  of  1000  men, 
and,  increasing  each  side  of  the  square  by  5  men,  lacked  25  men 
to  complete  the  square  ;  how  many  men  were  there  in  the  original 
army? 

48.  Find  the  cubic  inches  in  a  pail  12  in.  deep,  16  in.  wide  at 
top  and  12  in.  at  bottom. 

49.  A  life  annuity  costs  a  person  44  yrs.  old  $5933.35.    Find  the 
amount  of  the  annuity,  interest  at  3%%. 

50.  A  person  investing  in  a  4%  stock  receives  4f  %  for  his 
money.     What  is  the  price  of  the  stock  ? 

5  1.  A  piece  of  work  is  to  be  completed  in  30  days,  and  15  men 
are  employed  upon  it  ;  at  the  end  of  24  days  the  work  is  only  half 
done.  How  many  more  men  must  be  employed  to  fulfil  the  con- 
tract? 

52.  A  man  buys  eggs  at  a  certain  price  per  score,  and  sells  them 
at  half  that  price  per  dozen.    What  is  his  gain  or  loss  per  cent.? 

53.  A.'s  present  age  is  to  B.'s  as  9  is  to  5  ;  three  years  ago  the 
proportion  was  10  to  3.    Find  the  present  age  of  each. 

54.  Simplify  If  of  ||  of  ||| 


55.  Simplify 


54 


434 


PRACTICAL  ARITHMETIC 


Some  Commercial  Laws  Tabulated. 


STATES  AND  TERRITORIES. 

GRACE 
ALLOWED. 

LEGAL 
RATE  OF 
INTEREST. 

P 

"l 
sg 

PENALTY  FOE  USURY. 

ARREST 
FOR  DEBT. 

i 

Alabama  .  . 

Yes 

8<f<, 

Forfeit  interest 

No 

Arizona 

Yes 

7? 

No* 

Arkansas  
California  
Colorado  
Connecticut 

Yes 

No!! 
No 
No  II 

n 

8$ 
6$ 

fo* 

Any  rate 

Any  rate 

Forfeit  claim 

None 
None 

No* 
No* 
No* 
No* 

Delaware  
District  of  Columbia  .  .  . 
Florida  

Yes 
No 
No 

6# 

11 

Any  rate 
100 

10$ 

Forfeit  principal 
Forfeit  interest 
Forfeit  interest 

No* 
No 
No 

Georgia  
Idaho  :  .  .  
Illinois  
Indiana  
Indian  Territory  .... 
Iowa 

Yes 
No|| 
No|| 
Yes 
Yes 
Yes 

\\ 

5$ 
6$ 
6$ 
6$ 

8$ 
12$ 
7$ 
8$ 
10$ 
84 

Forfeit  excess  of  int. 
Forfeit  int.  &  10$  of  pr. 
Forfeit  interest 
Forfeit  excess  of  int. 
Void  as  to  prin.  and  int. 

No 

No* 
No* 
No* 

No* 

Kansas  
Kentucky  
Louisiana 

Yes 
Yes 
Yes 

6$ 
6$ 

5$ 

10$ 

8$ 

Forfeit  interest  over  10$ 
Forfeit  interest 
Forfeit  interest 

No* 
No* 

Not 

Maine  

No 

6$ 

Any  rate 

None 

Yes 

Maryland  •  .  . 

No 

6$ 

Forfeit  excess  of  int 

No 

Massachusetts  
Michigan 

No 
Yes 

6$ 

6$ 

Any  rate 

8$ 

None 
Forfeit  excess  of  int. 

Yes 

Not 

Minnesota 

Yes 

7* 

10$ 

None 

No 

Mississippi  
Missouri  
Montana  

Yes 
Yes 

No|| 

6$ 
6$ 
8$ 

10$ 
8$ 
Any  rate 

Forfeit  interest 
Forfeit  interest 
None 

No 
No 
No* 

Nebraska  .  . 

\es 

5$ 

Io$ 

Forfeit  interest 

No* 

Nevada  

Yes 

7$ 

Any  rate 

None 

Yes 

New  Hampshire  
New  Jersey  
New  Mexico  
New  York  

No 

No|| 
Yes 

No|| 

6$ 
6$ 
6$ 
6$ 

12$ 

Forft  3  times  exc.  &  costs 
Forfeit  interest 
Forfeit  interest 
Forfeit  claim 

Yes 

No*g 
Yes 
No* 

North  Carolina  
North  Dakota  
Ohio  
Oklahoma 

Yes 

No|| 
No 
Yes 

6$ 
7* 

6$ 
7$ 

12$ 
8$ 
12$ 

Forfeit  interest 
Forfeit  interest 
Forfeit  excess  of  8$ 

Yes 
No* 
No* 

Oregon  
Pennsylvania  
Rhode  Island  
South  Carolina  
South  Dakota 

No|| 
No|| 
Yes 
Yes 
Yes 

8% 
6$ 

6$ 

n 

10$ 

Any  rate 

8$ 
12$ 

Forfeit  claim 
Forfeit  excess  of  int. 
None 
Forfeit  interest 
Forfeit  interest 

No* 
No* 
Yesg 
Yes 
No* 

Tennessee  
Texas  
Utah 

Yes 
Yes 

No  || 

\\ 

8% 

10$ 
Any  rate 

Forfeit  excess  of  int. 
Forfeit  interest 
None 

No 
No 
Yes 

Vermont  
Virginia  
Washington  
West  Virginia  
Wisconsin  •.  . 
Wyoming  

No|| 
Yes 
Yes 
Yes 

No|| 
Yes 

It 

% 

11 

12$ 
10$ 

Forfeit  excess  of  int. 
Forfeit  interest 
None 
Forfeit  excess  of  int. 
Forfeit  interest 
None 

Yesg 
Yes 
Yes 
Yes 
Yes 
Yes? 

*  Except  in  cases  of  fraud.  J  Except  in  cases  of  fraud  and  breaches  of  trust 

f  Except  to  secure  person  of  debtor  to  answer  suit,          $  Except  females. 
||  Forbidden  absolutely. 


ANSWERS   TO   APPENDIX. 


DUODECIMALS. 
Page  4OO. 

1.  65  ft.  11'  3". 

2.  24ft.  8'  8"  10"". 

3.  693  ft.  11'  2"  8'". 

4.  22  ft.  V  3"  7'"  10"". 

5.  640  ft.  8'  5"  10'"  11"". 

6.  6  ft.  V  1"  9'". 

7.  7  ft.  10'  0"  6'"  9"". 

8.  147  ft.  9'  2"  3"'. 

9.  107  ft.  6'  4"  9"'  2"". 
10.  11  ft.  10'  7". 

Page  401. 

1.  17  ft.  9'  6"  II"7  8"". 

2.  448  cu.  ft.  7'  4". 

3.  927  sq.  ft.  3'  4". 

4.  74.7  cords. 

5.  30  ft.  2'  10"  2"'. 

6.  $5.55^ 

7.  452£ff  loads. 

1.  20  ft.  V  5". 

2.  5  ft.  7'  4"  3"',  etc. 

3.  20  ft.  3'. 

4.  $21.00. 


Page  4O2. 


6.  1  ft.  8', 


METRIC    SYSTEM. 
Page  4O7. 

1.  57654000mm;  5765. 4dkm; 

5765400cm;  576.54hm; 
576540dm;  57.654km. 

2.  261.914m. 

3.  312. 2m. 

4.  3600<*m;  .36ihm;  .0036km. 

5.  $294. 

6.  304.72a. 

7.  57cdm- 

8.  321148.8. 

9.  23.4st. 

10.  600hl. 

11.  540hl. 

12.  10"  or  10001. 

13.  3000000s;   3,000,000,000"*; 

SOOOOO1**;  30000000d«. 

14.  7068. 6ks. 

15.  17999.982™. 

16.  1440. 

17.  .08125a. 

18.  He  loses  2ff%. 

19.  $39.273. 

20.  .800m. 

21.  1932.69  sq.  in.  ;  7989.46  cu.  in. 

22.  Sea  water  102  6**;  milk  103.2*s; 

sea-water    225.72    lb.;   milk 
227.04  lb. 

435 


436 


ANSWERS  TO  APPENDIX 


Page  4O8. 

Page  419. 

23.  3com  ;  T35  cu.  in. 

10.   103;   2704.       20.   5  yrs. 

24.  8250  yd.  =  4^  mi. 

11.  486.                   21.  $14771.75. 

25.  1.69ha. 

12.  £.                      22.   $20715.436. 

26.  2Ty. 

13.  fa.                   23.  $18343.62. 

27.  $2.79. 

14.  58.                     24.  $63131.22. 

28.  16104.176m. 

15.  110;  630.          25.  $36001.188. 

29.  905071. 

16.  2048.                 26.  $500. 

30.  7.53251. 

17.  $1.50073.          27.  $360. 

18.  $3378.96.          28.  $968.45. 

FOREIGN   EXCHANGE. 

19.  $209707.90      29.  $7416.39. 

Page  410. 

Page  421. 

1.  $5085.49£. 
2.  $101939832.187. 
3.  792590673.575  pesetas  ; 
$51030167.813. 

i.  i.      11.  Hit-    21.  mi 

2.  f.          12.  T9r.            22.  ||  ||. 
3.  f.           13.  Iff.          23.  if 
4.  T4..         14.  TiT6T.          24.  3//5. 

4.  $476. 

5.  /T.        15.  fff.          25.  j|. 

5.  2433.25. 

£         2                "jp           1                     O£      0  1  0  4 

6.  $3851.43. 

•i.  HI.      17.  jifc.        27.  |f|!' 

Q        8                  1  Q         911                OQ      £137 

Page  411. 

Q        59              -JO           ]                     29      i"jT 

7.  2746.  725  francs. 

10.  ST-    20.  AWrV    30.  4Hff 

8.  $3260.60. 

9.  $5.73. 

Page  422. 

10.  $3625.45. 

II1               3s               5      l 

11.  $2436.29. 

2    *»'             4'  "'             6    ™'. 

12.  $540.40. 

¥T1T'                  '    ?'                       '    WV 

13.  $1547.00. 

2.  1.   168.           3.  -ip,           5.  95. 

14.  $1254.286. 

2.  -6^.             4.  693.           6.  ifA. 

15.  $2894.43. 

Page  423. 

ARITHMETICAL   PRO- 
GRESSION. 

5.  C.,  0°;  K.,0°. 
6.  P.,  122°;  K.,  40°. 
7.  F.,  —4°;  K.,  —16°. 

Page  418. 

2.  1.  32T8T  min.  past  6. 

1.  36.                       6.  645. 

2.  43T7T  min.  past  8. 

2.  134.                     7.  25£;  1287£. 

3.  5^T  min.  past  1. 

3.  57.                      8.  2316. 

4.  16^T  min.  past  3. 

4.  4.                         9.  20200. 

6.  49^T  min.  past  9. 

5.  156. 

6.  27^T  min.  past  5. 

ANSWERS   TO   APPENDIX 


437 


Page  424. 

35.  6^fV             48.  1. 

3.  1.  32T8r  min.  past  12. 

36.  .1.                  49.  10. 

2.  49^  min.  past  3. 

37.  ^f.                 50.  |f 

3.  16T4T  min.  past  9. 

38.  20^|.             51.  3|. 

4.  27T3T  min.  past  11. 
4.  1.  27T3T  min.  past  2. 

39.  0.                   52.  A. 
40.   15^.             53.  2A. 
41.  .09568.           54.  $2.50. 

2.  5^T  min.  past  4. 
3.  32T8T  min.  past  9. 
4.  43j2j-  min.  ;  ll^y  min.  past  11. 

42.  f$f  .               55.  $56.25. 
43.  1¥V-               56.  1. 
44.  655.36.          57.  .000000002035. 

5.  5T5T  min.  past  2. 

45.  l|f               58.   2f 

3.  l&days. 

46.  7.                   59.   |f 

47.  f 

Page  425. 

4.  4|£days. 
5.  T6j  of  an  hour. 

Page  429. 

6.   2|  min. 

60.  5^.                   77.  f 

7.  A.,  20  da.  ;  B.,  30  da.  ;  C.,  60  da. 

61.  if-                       78.  |f. 

8.  B.,  15  da. 

62.  m-                    79.  ft/,. 

9.  B.,  6  da.  ;  A.  and  C.,  eacli  12  da. 

63.  1.11.                   80.  sW*. 

10.  4  hr.  33  min. 

64.  V.                       81.  2.8665. 

65.  lyd.lft.5fin.  82.  f 

Page  427. 

66.  ff.                       83.  3Jf 

3.  29  da.  fr.  Apr.  10.      4.  $1198.60. 

67.  ^V                     84.  |f. 
68.  |f.                       85.  .0575. 

1.  132.             8.  f.                  15.  f 

69.  9.                         86.  .02847  -f  . 

2.  4f              9.  1|§.             16.  6f 

70.  24.                       87.  3s.  8d.  TY&. 

3.  4?VW-       10.  5H.             17.  Itf. 

71.  T5A.                     88.  42. 

4        ^                         1  1         ^                             1  fi      9  ^  3 

72.  1A.                     89.  T^. 

5.  12.71.        12.  lf£f.           19.  2|. 

73.  3.848.                  90.  .98—. 

6.  6|.            13.  I                20.  ^|. 

74.  l^V                     91.  .238+. 

7-  yMU.       14.  f 

75.  2.7081                 92.  .058302. 

76.  99  r8T. 

Page  428. 

21.  2f££.         28.  16|f 

22.  13Ty         29.  f 

Page  43O. 

23.   1.                30.  1.571428. 

93.  ^.                    98.  9^1- 

24.  T\V           31.  -mf. 

94.  54.0000658.      99.   1. 

25.  18ff.         32.  If 

95.   1.                     100.  2.068501. 

26.  .0545.        33.  7ff 

96.   1.                      101.  Ifff. 

27.  ft.             34.  A. 

97.  HI. 

THIS  BOOK  IS  DUB  ON  THBLAST  DATE 
STAMPED  BELOW 


^fT^r^ssssr^ 

OVERDUE. 


131935 
SEP  131935 


YB  17423 


8<X>553 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


